Large language models (LLMs) often generate convincing, fluent explanations. However, different from humans, they often generate inconsistent explanations on different inputs. For example, an LLM may explain “all birds can fly” when answering the question “Can sparrows fly?” but meanwhile answer “no” to the related question “Can penguins fly?”. Explanations should be consistent across related examples so that they allow humans to simulate the LLM’s decision process on multiple examples. We propose explanation-consistency finetuning (EC-finetuning), a method that adapts LLMs to generate more consistent natural-language explanations on related examples. EC-finetuning involves finetuning LLMs on synthetic data that is carefully constructed to contain consistent explanations. Across a variety of question-answering datasets in various domains, EC-finetuning yields a 10.0% relative explanation consistency improvement on 4 finetuning datasets, and generalizes to 7 out-of-distribution datasets not seen during finetuning (+4.5% relative). We will make our code available for reproducibility.

Automated mechanistic interpretation research has attracted great interest due to its potential to scale explanations of neural network internals to large models. Existing automated circuit discovery work relies on activation patching or its approximations to identify subgraphs in models for specific tasks (circuits). They often suffer from slow runtime, approximation errors, and specific requirements of metrics, such as non-zero gradients. In this work, we introduce contextual decomposition for transformers (CD-T) to build interpretable circuits in large language models. CD-T can produce circuits at any level of abstraction and is the first to efficiently produce circuits as fine-grained as attention heads at specific sequence positions. CD-T is compatible to all transformer types, and requires no training or manually-crafted examples. CD-T consists of a set of mathematical equations to isolate contribution of model features. Through recursively computing contribution of all nodes in a computational graph of a model using CD-T followed by pruning, we are able to reduce circuit discovery runtime from hours to seconds compared to state-of-the-art baselines. On three standard circuit evaluation datasets (indirect object identification, greater-than comparisons, and docstring completion), we demonstrate that CD-T outperforms ACDC and EAP by better recovering the manual circuits with an average of 97% ROC AUC under low runtimes. In addition, we provide evidence that faithfulness of CD-T circuits is not due to random chance by showing our circuits are 80% more faithful than random circuits of up to 60% of the original model size. Finally, we show CD-T circuits are able to perfectly replicate original models' behavior(faithfulness = 1) using fewer nodes than the baselines for all tasks. Our results underscore the great promise of CD-T for efficient automated mechanistic interpretability, paving the way for new insights into the workings of large language models.

Large language models (LLMs) have revolutionized machine learning due to their ability to capture complex interactions between input features. Popular post-hoc explanation methods like SHAP provide marginal feature attributions, while their extensions to interaction importances only scale to small input lengths ($\approx 20$). We propose Spectral Explainer (SPEX), a model-agnostic interaction attribution algorithm that efficiently scales to large input lengths ($\approx 1000)$. SPEX exploits underlying natural sparsity among interactions—common in real-world data—and applies a sparse Fourier transform using a channel decoding algorithm to efficiently identify important interactions. We perform experiments across three difficult long-context datasets that require LLMs to utilize interactions between inputs to complete the task. For large inputs, SPEX outperforms marginal attribution methods by up to 20% in terms of faithfully reconstructing LLM outputs. Further, SPEX successfully identifies key features and interactions that strongly influence model output. For one of our datasets, HotpotQA, SPEX provides interactions that align with human annotations. Finally, we use our model-agnostic approach to generate explanations to demonstrate abstract reasoning in closed-source LLMs (GPT-4o mini) and compositional reasoning in vision-language models.

We develop Latent Exploration Score (LES) to mitigate over-exploration in Latent Space Optimization (LSO), a popular method for solving black-box discrete optimization problems. LSO utilizes continuous optimization within the latent space of a Variational Autoencoder (VAE) and is known to be susceptible to over-exploration, which manifests in unrealistic solutions that reduce its practicality. LES leverages the trained decoder’s approximation of the data distribution, and can be employed with any VAE decoder–including pretrained ones–without additional training, architectural changes or access to the training data. Our evaluation across five LSO benchmark tasks and twenty-two VAE models demonstrates that LES always enhances the quality of the solutions while maintaining high objective values, leading to improvements over existing solutions in most cases. We believe that new avenues to LSO will be opened by LES’ ability to identify out of distribution areas, differentiability, and computational tractability.

Large Language Models (LLMs) have achieved remarkable performance by capturing complex interactions between input features. To identify these interactions, most existing approaches require enumerating all possible combinations of features up to a given order, causing them to scale poorly with the number of inputs $n$. Recently, Kang et al. (2025) proposed SPEX, an information-theoretic approach that uses interaction sparsity to scale to $n \approx 10^3$ features. SPEX greatly improves upon prior methods but requires tens of thousands of model inferences, which can be prohibitive for large models. In this paper, we observe that LLM feature interactions are often *hierarchical*—higher-order interactions are accompanied by their lower-order subsets—which enables more efficient discovery. To exploit this hierarchy, we propose ProxySPEX, an interaction attribution algorithm that first fits gradient boosted trees to masked LLM outputs and then extracts the important interactions. Experiments across four challenging high-dimensional datasets show that ProxySPEX more faithfully reconstructs LLM outputs by 20\% over marginal attribution approaches while using *$10\times$ fewer inferences* than SPEX. By accounting for interactions, ProxySPEX efficiently identifies the most influential features, providing a scalable approximation of their Shapley values. Further, we apply ProxySPEX to two interpretability tasks. *Data attribution*, where we identify interactions among CIFAR-10 training samples that influence test predictions, and *mechanistic interpretability*, where we uncover interactions between attention heads, both within and across layers, on a question-answering task. The ProxySPEX algorithm is available at <https://github.com/mmschlk/shapiq>.

In this work, we investigate the dynamics of stochastic gradient descent (SGD) when training a single-neuron autoencoder with linear or ReLU activation on orthogonal data. We show that for this non-convex problem, randomly initialized SGD with a constant step size successfully finds a global minimum for any batch size choice. However, the particular global minimum found depends upon the batch size. In the full-batch setting, we show that the solution is dense (i.e., not sparse) and is highly aligned with its initialized direction, showing that relatively little feature learning occurs. On the other hand, for any batch size strictly smaller than the number of samples, SGD finds a global minimum that is sparse and nearly orthogonal to its initialization, showing that the randomness of stochastic gradients induces a qualitatively different type of "feature selection" in this setting. Moreover, if we measure the sharpness of the minimum by the trace of the Hessian, the minima found with full-batch gradient descent are flatter than those found with strictly smaller batch sizes, in contrast to previous works which suggest that large batches lead to sharper minima. To prove convergence of SGD with a constant step size, we introduce a powerful tool from the theory of non-homogeneous random walks which may be of independent interest.

Many statistical estimators are defined as the fixed point of a data-dependent operator, with estimators based on minimizing a cost function being an important special case. The limiting performance of such estimators depends on the properties of the population-level operator in the idealized limit of infinitely many samples. We develop a general framework that yields bounds on statistical accuracy based on the interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (in)stability when applied to an empirical object based on $n$ samples. Using this framework, we analyze both stable forms of gradient descent and some higher-order and unstable algorithms, including Newton's method and its cubic-regularized variant, as well as the EM algorithm. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models. We exhibit cases in which an unstable algorithm can achieve the same statistical accuracy as a stable algorithm in exponentially fewer steps---namely, with the number of iterations being reduced from polynomial to logarithmic in sample size $n$.

Domain adaptation (DA) is a statistical learning problem that arises when the distribution of the source data used to train a model differs from that of the target data used to evaluate the model. While many DA algorithms have demonstrated considerable empirical success, blindly applying these algorithms can often lead to worse performance on new datasets. To address this, it is crucial to clarify the assumptions under which a DA algorithm has good target performance. In this work, we focus on the assumption of the presence of conditionally invariant components (CICs), which are relevant for prediction and remain conditionally invariant across the source and target data. We demonstrate that CICs, which can be estimated through conditional invariant penalty (CIP), play three prominent roles in providing target risk guarantees in DA. First, we propose a new algorithm based on CICs, importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings such as covariate shift and label shift. Second, we show that CICs help identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm can address its failure scenario caused by label-flipping features. We support our new algorithms and theoretical findings via numerical experiments on synthetic data, MNIST, CelebA, Camelyon17, and DomainNet datasets.

Pre-trained transformers are often fine-tuned to aid clinical decision-making using limited clinical notes. Model interpretability is crucial, especially in high-stakes domains like medicine, to establish trust and ensure safety, which requires human engagement. We introduce SUFO, a systematic framework that enhances interpretability of fine-tuned transformer feature spaces. SUFO utilizes a range of analytic and visualization techniques, including Supervised probing, Unsupervised similarity analysis, Feature dynamics, and Outlier analysis to address key questions about model trust and interpretability (e.g. model suitability for a task, feature space evolution during fine-tuning, and interpretation of fine-tuned features and failure modes). We conduct a case study investigating the impact of pre-training data where we focus on real-world pathology classification tasks, and validate our findings on MedNLI. We evaluate five 110M-sized pre-trained transformer models, categorized into general-domain (BERT, TNLR), mixed-domain (BioBERT, Clinical BioBERT), and domain-specific (PubMedBERT) groups. Our SUFO analyses reveal that: (1) while PubMedBERT, the domain-specific model, contains valuable information for fine-tuning, it can overfit to minority classes when class imbalances exist. In contrast, mixed-domain models exhibit greater resistance to overfitting, suggesting potential improvements in domain-specific model robustness; (2) in-domain pre-training accelerates feature disambiguation during fine-tuning; and (3) feature spaces undergo significant sparsification during this process, enabling clinicians to identify common outlier modes among fine-tuned models as demonstrated in this paper. These findings showcase the utility of SUFO in enhancing trust and safety when using transformers in medicine, and we believe SUFO can aid practitioners in evaluating fine-tuned language models (LMs) for other applications in medicine and in more critical domains.

In human-written articles, we often leverage the subtleties of text style, such as bold and italics, to guide the attention of readers. These textual emphases are vital for the readers to grasp the conveyed information. When interacting with large language models (LLMs), we have a similar need -- steering the model to pay closer attention to user-specified information, e.g., an instruction. Existing methods, however, are constrained to process plain text and do not support such a mechanism. This motivates us to introduce PASTA -- Post-hoc Attention STeering Approach, a method that allows LLMs to read text with user-specified emphasis marks. To this end, PASTA identifies a small subset of attention heads and applies precise attention reweighting on them, directing the model attention to user-specified parts. Like prompting, PASTA is applied at inference time and does not require changing any model parameters. Experiments demonstrate that PASTA can substantially enhance an LLM's ability to follow user instructions or integrate new knowledge from user inputs, leading to a significant performance improvement on a variety of tasks, e.g., an average accuracy improvement of 22\% for LLAMA-7B. Our code is publicly available at https://github.com/QingruZhang/PASTA .

In this paper, we show that Low Rank Adaptation (LoRA) as originally introduced in (Hu et al., 2021) leads to suboptimal finetuning of models with large width. This is due to the fact that adapter matrices A and B in LoRA are updated with the same learning rate in ADAM. Using scaling arguments for large width networks, we demonstrate that the same learning rate does not allow efficient feature learning. We then show that this suboptimality of LoRA can be corrected simply by setting different learning rates for the LoRA adapter matrices A and B with a well-chosen fixed ratio. We call this proposed algorithm LoRA+. In our extensive experiments, LoRA+ improves finetuning speed (up to ∼ 2X SpeedUp) and performance (1% − 2% improvements), at the same computational cost as LoRA. The code is available at https://github.com/nikhil-ghosh-berkeley/loraplus

In recent years, work has gone into developing deep interpretable methods for image classification that clearly attributes a model’s output to specific features of the data. One such of these methods is the Prototypical Part Network (ProtoPNet), which attempts to classify images based on meaningful parts of the input. While this architecture is able to produce visually interpretable classifications, it often learns to classify based on parts of the image that are not semantically meaningful. To address this problem, we propose the Reward Reweighing, Reselecting, and Retraining (R3) post-processing framework, which performs three additional corrective updates to a pretrained ProtoPNet in an offline and efficient manner. The first two steps involve learning a reward model based on collected human feedback and then aligning the prototypes with human preferences. The final step is retraining, which realigns the base features and the classifier layer of the original model with the updated prototypes. We find that our R3 framework consistently improves both the interpretability and the predictive accuracy of ProtoPNet and its variants.

Transfer learning is a critical part of real-world machine learning deployments and has been extensively studied in experimental works with overparameterized neural networks. However, even in the simplest setting of linear regression a notable gap still exists in the theoretical understanding of transfer learning. In-distribution research on high-dimensional linear regression has led to the identification of a phenomenon known as benign overfitting, in which linear interpolators overfit to noisy training labels and yet still generalize well. This behavior occurs under specific conditions on the source covariance matrix and input data dimension. Therefore, it is natural to wonder how such high-dimensional linear models behave under transfer learning. We prove the first non-asymptotic excess risk bounds for benignly-overfit linear interpolators in the transfer learning setting. From our analysis, we propose a taxonomy of beneficial and malignant covariate shifts based on the degree of overparameterization. We follow our analysis with empirical studies that show these beneficial and malignant covariate shifts for linear interpolators on real image data, and for fully-connected neural networks in settings where the input data dimension is larger than the training sample size.

In the emergency department (ED), patients undergo triage and multiple laboratory tests before diagnosis. This time-consuming process causes ED crowding which impacts patient mortality, medical errors, staff burnout, etc. This work proposes (time) cost-effective diagnostic assistance that leverages artificial intelligence systems to help ED clinicians make efficient and accurate diagnoses. In collaboration with ED clinicians, we use public patient data to curate MIMIC-ED-Assist, a benchmark for AI systems to suggest laboratory tests that minimize wait time while accurately predicting critical outcomes such as death. With MIMIC-ED-Assist, we develop ED-Copilot which sequentially suggests patient-specific laboratory tests and makes diagnostic predictions. ED-Copilot employs a pre-trained bio-medical language model to encode patient information and uses reinforcement learning to minimize ED wait time and maximize prediction accuracy. On MIMIC-ED-Assist, ED-Copilot improves prediction accuracy over baselines while halving average wait time from four hours to two hours. ED-Copilot can also effectively personalize treatment recommendations based on patient severity, further highlighting its potential as a diagnostic assistant. Since MIMIC-ED-Assist is a retrospective benchmark, ED-Copilot is restricted to recommend only observed tests. We show ED-Copilot achieves competitive performance without this restriction as the maximum allowed time increases. Our code is available at https://github.com/cxcscmu/ED-Copilot.

In this paper, we study the role of initialization in Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021). Essentially, to start from the pretrained model, one can either initialize $B$ to zero and $A$ to random, or vice-versa. In both cases, the product $BA$ is equal to zero at initialization, which makes finetuning starts from the pretrained model. These two initialization schemes are seemingly similar. They should in-principle yield the same performance and share the same optimal learning rate. We demonstrate that this is an *incorrect intuition* and that the first scheme (of initializing $B$ to zero and $A$ to random) on average in our experiments yields better performance compared to the other scheme. Our theoretical analysis shows that the reason behind this might be that the first initialization allows the use of larger learning rates (without causing output instability) compared to the second initialization, resulting in more efficient learning of the first scheme. We validate our results with extensive experiments on LLMs.

Backpropagation, the cornerstone of deep learning, is limited to computing gradients for continuous variables. This limitation poses challenges for problems involving discrete latent variables. To address this issue, we propose a novel approach to approximate the gradient of parameters involved in generating discrete latent variables. First, we examine the widely used Straight-Through (ST) heuristic and demonstrate that it works as a first-order approximation of the gradient. Guided by our findings, we propose ReinMax, which achieves second-order accuracy by integrating Heun’s method, a second-order numerical method for solving ODEs. ReinMax does not require Hessian or other second-order derivatives, thus having negligible computation overheads. Extensive experimental results on various tasks demonstrate the superiority of ReinMax over the state of the art.

Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen's principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for overparameterized models. MDL-COMP is defined via an optimality criterion over the encodings induced by a good Ridge estimator class. We provide an extensive theoretical characterization of MDL-COMP for linear models and kernel methods and show that it is not just a function of parameter count, but rather a function of the singular values of the design or the kernel matrix and the signal-to-noise ratio. For a linear model with $n$ observations, $d$ parameters, and i.i.d. Gaussian predictors, MDL-COMP scales linearly with $d$ when $dn$. For kernel methods, we show that MDL-COMP informs minimax in-sample error, and can decrease as the dimensionality of the input increases. We also prove that MDL-COMP upper bounds the in-sample mean squared error (MSE). Via an array of simulations and real-data experiments, we show that a data-driven Prac-MDL-COMP informs hyper-parameter tuning for optimizing test MSE with ridge regression in limited data settings, sometimes improving upon cross-validation and (always) saving computational costs. Finally, our findings also suggest that the recently observed double decent phenomenons in overparameterized models might be a consequence of the choice of non-ideal estimators. [abs][pdf][bib] [code] © JMLR 2023. (edit, beta) Mastodon

Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.

Decision trees and random forests (RF) are a cornerstone of modern machine learning practice. Due to their tendency to overfit, trees are typically regularized by a variety of techniques that modify their structure (e.g. pruning). We introduce Hierarchical Shrinkage (HS), a post-hoc algorithm which regularizes the tree not by altering its structure, but by shrinking the prediction over each leaf toward the sample means over each of its ancestors, with weights depending on a single regularization parameter and the number of samples in each ancestor. Since HS is a post-hoc method, it is extremely fast, compatible with any tree-growing algorithm and can be used synergistically with other regularization techniques. Extensive experiments over a wide variety of real-world datasets show that HS substantially increases the predictive performance of decision trees even when used in conjunction with other regularization techniques. Moreover, we find that applying HS to individual trees in a RF often improves its accuracy and interpretability by simplifying and stabilizing decision boundaries and SHAP values. We further explain HS by showing that it to be equivalent to ridge regression on a basis that is constructed of decision stumps associated to the internal nodes of a tree. All code and models are released in a full-fledged package available on Github

We study the problem of learning robust acoustic models in adverse environments, characterized by a significant mismatch between training and test conditions. This problem is of paramount importance for the deployment of speech recognition systems that need to perform well in unseen environments. First, we characterize data augmentation theoretically as an instance of vicinal risk minimization, which aims at improving risk estimates during training by replacing the delta functions that define the empirical density over the input space with an approximation of the marginal population density in the vicinity of the training samples. More specifically, we assume that local neighborhoods centered at training samples can be approximated using a mixture of Gaussians, and demonstrate theoretically that this can incorporate robust inductive bias into the learning process. We then specify the individual mixture components implicitly via data augmentation schemes, designed to address common sources of spurious correlations in acoustic models. To avoid potential confounding effects on robustness due to information loss, which has been associated with standard feature extraction techniques (e.g.,fbankandmfccfeatures), we focus on the waveform-based setting. Our empirical results show that the approach can generalize to unseen noise conditions, with 150% relative improvement in out-of-distribution generalization compared to training using the standard risk minimization principle. Moreover, the results demonstrate competitive performance relative to models learned using a training sample designed to match the acoustic conditions characteristic of test utterances.

Recent deep-learning models have achieved impressive prediction performance, but often sacrifice interpretability and computational efficiency. Interpretability is crucial in many disciplines, such as science and medicine, where models must be carefully vetted or where interpretation is the goal itself. Moreover, interpretable models are concise and often yield computational efficiency. Here, we propose adaptive wavelet distillation (AWD), a method which aims to distill information from a trained neural network into a wavelet transform. Specifically, AWD penalizes feature attributions of a neural network in the wavelet domain to learn an effective multi-resolution wavelet transform. The resulting model is highly predictive, concise, computationally efficient, and has properties (such as a multi-scale structure) which make it easy to interpret. In close collaboration with domain experts, we showcase how AWD addresses challenges in two real-world settings: cosmological parameter inference and molecular-partner prediction. In both cases, AWD yields a scientifically interpretable and concise model which gives predictive performance better than state-of-the-art neural networks. Moreover, AWD identifies predictive features that are scientifically meaningful in the context of respective domains. All code and models are released in a full-fledged package available on Github.

We study a class of weakly identifiable location-scale mixture models for which the maximum likelihood estimates based on $n$ i.i.d. samples are known to have lower accuracy than the classical $n^{- \frac{1}{2}}$ error. We investigate whether the Expectation-Maximization (EM) algorithm also converges slowly for these models. We provide a rigorous characterization of EM for fitting a weakly identifiable Gaussian mixture in a univariate setting where we prove that the EM algorithm converges in order $n^{\frac{3}{4}}$ steps and returns estimates that are at a Euclidean distance of order ${ n^{- \frac{1}{8}}}$ and ${ n^{-\frac{1} {4}}}$ from the true location and scale parameter respectively. Establishing the slow rates in the univariate setting requires a novel localization argument with two stages, with each stage involving an epoch-based argument applied to a different surrogate EM operator at the population level. We demonstrate several multivariate ($d \geq 2$) examples that exhibit the same slow rates as the univariate case. We also prove slow statistical rates in higher dimensions in a special case, when the fitted covariance is constrained to be a multiple of identity.

For an explanation of a deep learning model to be effective, it must provide both insight into a model and suggest a corresponding action in order to achieve some objective. Too often, the litany of proposed explainable deep learning methods stop at the first step, providing practitioners with insight into a model, but no way to act on it. In this paper, we propose contextual decomposition explanation penalization (CDEP), a method which enables practitioners to leverage existing explanation methods to increase the predictive accuracy of a deep learning model. In particular, when shown that a model has incorrectly assigned importance to some features, CDEP enables practitioners to correct these errors by inserting domain knowledge into the model via explanations. We demonstrate the ability of CDEP to increase performance on an array of toy and real datasets.

Veridical data science extracts reliable and reproducible information from data, with an enriched technical language to communicate and evaluate empirical evidence in the context of human decisions and domain knowledge. Building and expanding on principles of statistics, machine learning, and the sciences, we propose the predictability, computability, and stability (PCS) framework forveridical data science. Our framework is comprised of both a workflow and documentation and aims to provide responsible, reliable, reproducible, and transparent results across the entire data science life cycle. Moreover, we propose the PDR desiderata for interpretable machine learning as part of veridical data science (with PDR standing for predictive accuracy, predictive accuracy and relevancy to a human audience and a particular domain problem).

We study the problem of globally recovering a dictionary from a set of signals via $\ell_1$-minimization. We assume that the signals are generated as i.i.d. random linear combinations of the $K$ atoms from a complete reference dictionary $D^*\in \mathbb R^{K\times K}$, where the linear combination coefficients are from either a Bernoulli type model or exact sparse model. First, we obtain a necessary and sufficient norm condition for the reference dictionary $D^*$ to be a sharp local minimum of the expected $\ell_1$ objective function. Our result substantially extends that of Wu and Yu (2015) and allows the combination coefficient to be non-negative. Secondly, we obtain an explicit bound on the region within which the objective value of the reference dictionary is minimal. Thirdly, we show that the reference dictionary is the unique sharp local minimum, thus establishing the first known global property of $\ell_1$-minimization dictionary learning. Motivated by the theoretical results, we introduce a perturbation based test to determine whether a dictionary is a sharp local minimum of the objective function. In addition, we also propose a new dictionary learning algorithm based on Block Coordinate Descent, called DL-BCD, which is guaranteed to decrease the obective function monotonically. Simulation studies show that DL-BCD has competitive performance in terms of recovery rate compared to other state-of-the-art dictionary learning algorithms when the reference dictionary is generated from random Gaussian matrices.

Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Stormer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of step-size and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algorithms such as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we provide a general framework for sharpening mixing time bounds of Markov chains initialized at a substantial distance from the target distribution over continuous spaces. We apply this sharpening device to the Metropolized random walk and Langevin algorithms, thereby obtaining improved mixing time bounds from a non-warm initial distribution.

Tree ensembles such as Random Forests have achieved impressive empirical success across a wide variety of applications. To understand how these models make predictions, people routinely turn to feature importance measures calculated from tree ensembles. It has long been known that Mean Decrease Impurity (MDI), one of the most widely used measures of feature importance, incorrectly assigns high importance to noisy features, leading to systematic bias in feature selection. In this paper, we address the feature selection bias of MDI from both theoretical and methodological perspectives. Based on the original definition of MDI by Breiman et al. \cite{Breiman1984} for a single tree, we derive a tight non-asymptotic bound on the expected bias of MDI importance of noisy features, showing that deep trees have higher (expected) feature selection bias than shallow ones. However, it is not clear how to reduce the bias of MDI using its existing analytical expression. We derive a new analytical expression for MDI, and based on this new expression, we are able to propose a debiased MDI feature importance measure using out-of-bag samples, called MDI-oob. For both the simulated data and a genomic ChIP dataset, MDI-oob achieves state-of-the-art performance in feature selection from Random Forests for both deep and shallow trees.

We study the problem of sampling from a strongly log-concave density supported on $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most $\delta$ for a density with condition number $\kappa$, we show that MALA requires $\mathcal{O} (\kappa d \log(1/\delta) )$ steps from a warm start, as compared to the $\mathcal{O} (\kappa^2 d/\delta^2 )$ steps established in past work on ULA. We also demonstrate the gains of a modified version of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for the Metropolized random walk (MRW) and obtain $\mathcal{O}(\kappa)$ mixing time slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.

In this talk, I'd like to discuss the intertwining importance and connections of three principles of data science in the title and the PCS workflow that is built on the three principles. The principles will be demonstrated in the context of two collaborative projects in neuroscience and genomics for interpretable data results and testable hypothesis generation.

We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by running a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step to ensure the correct stationary distribution. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds reveal that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most $\delta$ for a density with condition number $\kappa$, we show that MALA requires $\mathcal{O} \big(\kappa d \log(1/\delta) \big)$ steps, as compared to the $\mathcal{O} \big(\kappa^2 d/\delta^2 \big)$ steps established in past work on ULA. We also demonstrate the gains of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for a zeroth-order method Metropolized random walk (MRW) and show that it mixes $\mathcal{O}(\kappa d)$ slower than MALA.

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk mixes in significantly fewer steps than the logarithmic-barrier based Dikin walk studied in past work. For a polytope in $\mathbb{R}^d$ defined by $n > d$ linear constraints, we show that the mixing time from a warm start is bounded as $\mathcal{O}(n^{0.5}d^{1.5})$, compared to the $\mathcal{O}(nd)$ mixing time bound for the Dikin walk. The cost of each step of the Vaidya walk is of the same order as the Dikin walk, and at most twice as large in terms of constant pre-factors. For the John walk, we prove an $\mathcal{O}(d^{2.5}\cdot\log^4(n/d))$ bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of $\mathcal{O}(d^{2}\cdot\text{poly-log}(n/d))$. Additionally, we propose variants of the Vaidya and John walks that mix in polynomial time from a deterministic starting point. The speed-up of the Vaidya walk over the Dikin walk are illustrated in numerical examples.

Artificial intelligence (AI) is intrinsically data-driven. It calls for the application of statistical concepts through human-machine collaboration during the generation of data, the development of algorithms, and the evaluation of results. This paper discusses how such human-machine collaboration can be approached through the statistical concepts of population, question of interest, representativeness of training data, and scrutiny of results (PQRS). The PQRS workflow provides a conceptual framework for integrating statistical ideas with human input into AI products and researches. These ideas include experimental design principles of randomization and local control as well as the principle of stability to gain reproducibility and interpretability of algorithms and data results. We discuss the use of these principles in the contexts of self-driving cars, automated medical diagnoses, and examples from the authors’ collaborative research.

In this talk, I'd like to discuss the intertwining importance and connections of three principles of data science in the title in data-driven decisions.

Techniques for nonparametric regression based on fitting small-scale local models at prediction time have long been studied in statistics and pattern recognition, but have received less attention in modern large-scale machine learning applications. In practice, such methods are generally applied to low-dimensional problems, but may falter with high-dimensional predictors if they use a Euclidean distance-based kernel. We propose a new method, SILO, for fitting prediction-time local models that uses supervised neighborhoods that adapt to the local shape of the regression surface. To learn such neighborhoods, we use a weight function between points derived from random forests. We prove the consistency of SILO, and demonstrate through simulations and real data that our method works well in both the serial and distributed settings. In the latter case, SILO learns the weighting function in a divide-and-conquer manner, entirely avoiding communication at training time.

Several months ago, Phil Bourne, the initiator and frequent author of the wildly successful and incredibly useful “Ten Simple Rules” series, suggested that some statisticians put together a Ten Simple Rules article related to statistics. (One of the rules for writing a PLOS Ten Simple Rules article is to be Phil Bourne [1]. In lieu of that, we hope effusive praise for Phil will suffice.) Implicit in the guidelines for writing Ten Simple Rules [1] is “know your audience.” We developed our list of rules with researchers in mind: researchers having some knowledge of statistics, possibly with one or more statisticians available in their building, or possibly with a healthy do-it-yourself attitude and a handful of statistical packages on their laptops. We drew on our experience in both collaborative research and teaching, and, it must be said, from our frustration at being asked, more than once, to “take a quick look at my student’s thesis/my grant application/my referee’s report: it needs some input on the stats, but it should be pretty straightforward.” There are some outstanding resources available that explain many of these concepts clearly and in much more detail than we have been able to do here: among our favorites are Cox and Donnelly [2], Leek [3], Peng [4], Kass et al. [5], Tukey [6], and Yu [7]. Every article on statistics requires at least one caveat. Here is ours: we refer in this article to “science” as a convenient shorthand for investigations using data to study questions of interest. This includes social science, engineering, digital humanities, finance, and so on. Statisticians are not shy about reminding administrators that statistical science has an impact on nearly every part of almost all organizations.

One popular method for dealing with large-scale data sets is sampling. Using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. Existing work has focused on algorithmic issues, but none of it addresses statistical aspects of this method. Here, we provide an effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model. In particular, for several versions of leverage-based sampling, we derive results for the bias and variance, both conditional and unconditional on the observed data. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms: one constructs a smaller least-squares problem with “shrinked” leverage scores (SLEV), and the other solves a smaller and unweighted (or biased) least-squares problem (LEVUNW). The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.

Detection of clouds in satellite-generated radiance images, including those from MODIS, is an important first step in many applications of these data. In this paper we apply spectral unmixing to this problem with the aim of estimating subpixel cloud fractions, as opposed to identification only of whether or not a pixel radiance contains cloud contributions. We formulate the spectral unmixing approach in terms of multiple-kernel learning (MKL). To this end we propose a MKL-based unmixing algorithm that drives a multiple-kernel description of cloud, enabling estimation of sub-pixel cloud fractions. This approach is based on supervised learning. We generate training and testing samples by using CloudSat and CALIPSO data to compute cloud fractions within individual MODIS pixels. Results of our study on limited data (1875 training and testing MODIS pixels along with their CloudSat and CALIPSO based sub-pixel cloud fractions) show that the proposed algorithm can effectively estimate sub-pixel MODIS cloud fraction and outperforms support vector machine (SVM) in terms of estimation performance.

This paper proposes methods for combining cepstral and acoustic waveform representations for a front-end of support vector machine (SVM)-based speech recognition systems that are robust to additive noise. The key issue of kernel design and noise adaptation for the acoustic waveform representation is addressed first. Cepstral and acoustic waveform representations are then compared on a phoneme classification task. Experiments show that the cepstral features achieve very good performance in low noise conditions, but suffer severe performance degradation already at moderate noise levels. Classification in the acoustic waveform domain, on the other hand, is less accurate in low noise but exhibits a more robust behavior in high noise conditions. A combination of the cepstral and acoustic waveform representations achieves better classification performance than either of the individual representations over the entire range of noise levels tested, down to$-$18-dB SNR.

Consider the high-dimensional linear regression modely=Xβ*+w, wherey∈ \BBRnis an observation vector,X∈ \BBRn×dis a design matrix withd>;n, β*∈ \BBRdis an unknown regression vector, andw~N(0, σ2I) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimating β*in eitherl2-loss andl2-prediction loss, assuming that β*belongs to anlq-ball \BBBq(Rq) for someq∈ [0,1]. It is shown that under suitable regularity conditions on the design matrixX, the minimax optimal rate inl2-loss andl2-prediction loss scales as Θ(Rq([(logd)/(n)])1-q/2). The analysis in this paper reveals that conditions on the design matrixXenter into the rates forl2-error andl2-prediction error in complementary ways in the upper and lower bounds. Our proofs of the lower bounds are information theoretic in nature, based on Fano's inequality and results on the metric entropy of the balls \BBBq(Rq), whereas our proofs of the upper bounds are constructive, involving direct analysis of least squares overlq-balls. For the special caseq=0, corresponding to models with an exact sparsity constraint, our results show that although computationally efficientl1-based methods can achieve the minimax rates up to constant factors, they require slightly stronger assumptions on the design matrixXthan optimal algorithms involving least-squares over thel0-ball.

Multi-task learning aims at combining information across tasks to boost prediction performance, especially when the number of training samples is small and the number of predictors is very large. In this paper, we first extend the Sparse Discriminate Analysis (SDA) of Clemmensen et al.. We call this Multi-task Sparse Discriminate Analysis (MtSDA). MtSDA formulates multi-label prediction as a quadratic optimization problem whereas SDA obtains single labels via a nearest class mean rule. Second, we propose a class of equicorrelation matrices to use in MtSDA which includes the identity matrix. MtSDA with both matrices are compared with singletask learning (SVM and LDA+SVM) and multi-task learning (HSML). The comparisons are made on real data sets in terms of AUC and F-measure. The data results show that MtSDA outperforms other methods substantially almost all the time and in some cases MtSDA with the equicorrelation matrix substantially outperforms MtSDA with identity matrix.

Predicting the execution time of computer programs is an important but challenging problem in the community of computer systems. Existing methods require experts to perform detailed analysis of program code in order to construct predictors or select important features. We recently developed a new system to automatically extract a large number of features from program execution on sample inputs, on which prediction models can be constructed without expert knowledge. In this paper we study the construction of predictive models for this problem. We propose the SPORE (Sparse POlynomial REgression) methodology to build accurate prediction models of program performance using feature data collected from program execution on sample inputs. Our two SPORE algorithms are able to build relationships between responses (e.g., the execution time of a computer program) and features, and select a few from hundreds of the retrieved features to construct an explicitly sparse and non-linear model to predict the response variable. The compact and explicitly polynomial form of the estimated model could reveal important insights into the computer program (e.g., features and their non-linear combinations that dominate the execution time), enabling a better understanding of the program’s behavior. Our evaluation on three widely used computer programs shows that SPORE methods can give accurate prediction with relative error less than 7% by using a moderate number of training data samples. In addition, we compare SPORE algorithms to state-of-the-art sparse regression algorithms, and show that SPORE methods, motivated by real applications, outperform the other methods in terms of both interpretability and prediction accuracy.

Information theory provides an attractive framework for attacking the neural coding problem. This entails estimating information theoretic quantities from neural spike train data. This paper highlights two issues that may arise: non-parametric entropy estimation and non-stationarity. It gives an overview of these issues and some of the progress that has been made.

The estimation of high-dimensional parametric models requires imposing some structure on the models, for instance that they be sparse, or that matrix structured parameters have low rank. A general approach for such structured parametric model estimation is to use regularized M-estimation procedures, which regularize a loss function that measures goodness of fit of the parameters to the data with some regularization function that encourages the assumed structure. In this paper, we aim to provide a unified analysis of such regularized M-estimation procedures. In particular, we report the convergence rates of such estimators in any metric norm. Using just our main theorem, we are able to rederive some of the many existing results, but also obtain a wide range of novel convergence rates results. Our analysis also identifies key properties of loss and regularization functions such as restricted strong convexity, and decomposability, that ensure the corresponding regularized M-estimators have good convergence rates.

This paper uses information-theoretic techniques to determine minimax rates for estimating nonparametric sparse additive regression models under high-dimensional scaling. We assume an additive decomposition of the form $f^*(X_1, \ldots, X_p) = \sum_{j \in S} h_j(X_j)$, where each component function $h_j$ lies in some Hilbert Space $\Hilb$ and $S \subset \{1, \ldots, \pdim \}$ is an unknown subset with cardinality $\s = |S$. Given $\numobs$ i.i.d. observations of $f^*(X)$ corrupted with white Gaussian noise where the covariate vectors $(X_1, X_2, X_3,...,X_{\pdim})$ are drawn with i.i.d. components from some distribution $\mP$, we determine tight lower bounds on the minimax rate for estimating the regression function with respect to squared $\LTP$ error. The main result shows that the minimax rates are $\max{\big(\frac{\s \log \pdim / \s}{n}, \LowerRateSq \big)}$. The first term reflects the difficulty of performing \emph{subset selection} and is independent of the Hilbert space $\Hilb$; the second term $\LowerRateSq$ is an \emph{\s-dimensional estimation} term, depending only on the low dimension $\s$ but not the ambient dimension $\pdim$, that captures the difficulty of estimating a sum of $\s$ univariate functions in the Hilbert space $\Hilb$. As a special case, if $\Hilb$ corresponds to the $\m$-th order Sobolev space $\SobM$ of functions that are $m$-times differentiable, the $\s$-dimensional estimation term takes the form $\LowerRateSq \asymp \s \; n^{-2\m/(2\m+1)}$. The minimax rates are compared with rates achieved by an $\ell_1$-penalty based approach, it can be shown that a certain $\ell_1$-based approach achieves the minimax optimal rate.

Information estimates such as the direct method of Strong, Koberle, de Ruyter van Steveninck, and Bialek (1998) sidestep the difficult problem of estimating the joint distribution of response and stimulus by instead estimating the difference between the marginal and conditional entropies of the response. While this is an effective estimation strategy, it tempts the practitioner to ignore the role of the stimulus and the meaning of mutual information. We show here that as the number of trials increases indefinitely, the direct (or plug-in) estimate of marginal entropy converges (with probability 1) to the entropy of the time-averaged conditional distribution of the response, and the direct estimate of the conditional entropy converges to the time-averaged entropy of the conditional distribution of the response. Under joint stationarity and ergodicity of the response and stimulus, the difference of these quantities converges to the mutual information. When the stimulus is deterministic or nonstationary the direct estimate of information no longer estimates mutual information, which is no longer meaningful, but it remains a measure of variability of the response distribution across time.

In this paper we develop a spectral framework for estimating mixture distributions, specifically Gaussian mixture models. In physics, spectroscopy is often used for the identification of substances through their spectrum. Treating a kernel function K(x, y) as "light" and the sampled data as "substance", the spectrum of their interaction (eigenvalues and eigenvectors of the kernel matrix K) unveils certain aspects of the underlying parametric distribution p, such as the parameters of a Gaussian mixture. Our approach extends the intuitions and analyses underlying the existing spectral techniques, such as spectral clustering and Kernel Principal Components Analysis (KPCA).

We propose a novel hierarchical, nonlinear model that predicts brain activity in area V1 evoked by natural images. In the study reported here brain activity was measured by means of functional magnetic resonance imaging (fMRI), a noninvasive technique that provides an indirect measure of neural activity pooled over a small volume (~ 2mm cube) of brain tissue. Our model, which we call the SpAM V1 model, is based on the reasonable assumption that fMRI measurements reflect the (possibly nonlinearly) pooled, rectified output of a large population of simple and complex cells in V1. It has a hierarchical filtering stage that consists of three layers: model simple cells, model complex cells, and a third layer in which the complex cells are linearly pooled (called “pooled-complex” cells). The pooling stage then obtains the measured fMRI signals as a sparse additive model (SpAM) in which a sparse nonparametric (nonlinear) combination of model complex cell and model pooled-complex cell outputs are summed. Our results show that the SpAM V1 model predicts fMRI responses evoked by natural images better than a benchmark model that only provides linear pooling of model complex cells. Furthermore, the spatial receptive fields, frequency tuning and orientation tuning curves of the SpAM V1 model estimated for each voxel appears to be consistent with the known properties of V1, and with previous analyses of this data set. A visualization procedure applied to the SpAM V1 model shows that most of the nonlinear pooling consists of simple compressive or saturating nonlinearities.

Lasso is a regularization method for parameter estimation in linear models. It optimizes the model parameters with respect to a loss function subject to model complexities. This paper explores the use of lasso for statistical language modeling for text input. Owing to the very large number of parameters, directly optimizing the penalized lasso loss function is impossible. Therefore, we investigate two approximation methods, the boosted lasso (BLasso) and the forward stagewise linear regression (FSLR). Both methods, when used with the exponential loss function, bear strong resemblance to the boosting algorithm which has been used as a discriminative training method for language modeling. Evaluations on the task of Japanese text input show that BLasso is able to produce the best approximation to the lasso solution, and leads to a significant improvement, in terms of character error rate, over boosting and the traditional maximum likelihood estimation.

We propose Sparse Boosting (the SparseL2Boost algorithm), a variant on boosting with the squared error loss. SparseL2Boost yields sparser solutions than the previously proposed L2Boosting by minimizing some penalized L2-loss functions, the FPE model selection criteria, through small-step gradient descent. Although boosting may give already relatively sparse solutions, for example corresponding to the soft-thresholding estimator in orthogonal linear models, there is sometimes a desire for more sparseness to increase prediction accuracy and ability for better variable selection: such goals can be achieved with SparseL2Boost. We prove an equivalence of SparseL2Boost to Breiman's nonnegative garrote estimator for orthogonal linear models and demonstrate the generic nature of SparseL2Boost for nonparametric interaction modeling. For an automatic selection of the tuning parameter in SparseL2Boost we propose to employ the gMDL model selection criterion which can also be used for early stopping of L2Boosting. Consequently, we can select between SparseL2Boost and L2Boosting by comparing their gMDL scores.

Sparsity or parsimony of statistical models is crucial for their proper interpretations, as in sciences and social sciences. Model selection is a commonly used method to find such models, but usually involves a computationally heavy combinatorial search. Lasso (Tibshirani, 1996) is now being used as a computationally feasible alternative to model selection. Therefore it is important to study Lasso for model selection purposes. In this paper, we prove that a single condition, which we call the Irrepresentable Condition, is almost necessary and sufficient for Lasso to select the true model both in the classical fixed p setting and in the large p setting as the sample size n gets large. Based on these results, sufficient conditions that are verifiable in practice are given to relate to previous works and help applications of Lasso for feature selection and sparse representation. This Irrepresentable Condition, which depends mainly on the covariance of the predictor variables, states that Lasso selects the true model consistently if and (almost) only if the predictors that are not in the true model are "irrepresentable" (in a sense to be clarified) by predictors that are in the true model. Furthermore, simulations are carried out to provide insights and understanding of this result.

In this paper, a novel approach is proposed for estimating traffic matrices. Our method, called PamTram for PArtial Measurement of TRAffic Matrices, couples lightweight origin-destination (OD) flow measurements along with a computationally lightweight algorithm for producing OD estimates. The first key aspect of our method is to actively select a small number of informative OD flows to measure in each estimation interval. To avoid the heavy computation of optimal selection, we use intuition from game theory to develop randomized selection rules, with the goals of reducing errors and adapting to traffic changes. We show that it is sufficient to measure only one flow per measurement period to drastically reduce errors-thus rendering our method lightweight in terms of measurement overhead. The second key aspect is an explanation and proof that an Iterative Proportional Fitting algorithm approximates traffic matrix estimates when the goal is a minimum mean-squared error; this makes our method lightweight in terms of computation overhead. A one-step error bound is provided for PamTram that bounds the average error for the worst scenario. We validate our method using data from Sprint's European Tier-1 IP backbone network and demonstrate its consistent improvement over previous methods.

The convergence rates of generalized iterative scaling (GIS) and improved iterative scaling (IIS) algorithms for fitting maximum entropy (ME) models are investigated and also a particular linear dynamic system monitoring with partial active measurements is studied. An information-theoretic based measurement scheme is derived to select informative hidden states, which is validated on a problem of origin-destination matrix estimation for Internet traffic.

Network monitoring and diagnosis are key to improving network performance. The difficulties of performance monitoring lie in today's fast growing Internet, accompanied by increasingly heterogeneous and unregulated structures. Moreover, these tasks become even harder since one cannot rely on the collaboration of individual routers and servers to directly measure network traffic. Even though the aggregatory nature of possible network measurements gives rise to inverse problems, existing methods for solving inverse problems are usually computationally intractable or statistically inefficient. In this paper, a pseudo likelihood approach is proposed to solve a group of network tomography problems. The basic idea of pseudo likelihood is to form simple subproblems and construct a product of marginal likelihood of subproblems by the ignoring their dependences. As a result, it keeps a good balance between the computational complexity and the statistical efficiency of the parameter estimation. Some statistical properties of the pseudo likelihood estimator, such as consistency and asymptotic normality, are established. A pseudo expectation-maximization (EM) algorithm is developed to maximize the pseudo log-likelihood function. Two examples with simulated or real data are used to illustrate the pseudo likelihood proposal: (1) internal link delay distribution inference through multicast end-to-end measurements; (2) origin-destination matrix estimation through link traffic counts.

MOTIVATION: The microarray technology allows for the simultaneous monitoring of thousands of genes for each sample. The high-dimensional gene expression data can be used to study similarities of gene expression profiles across different samples to form a gene clustering. The clusters may be indicative of genetic pathways. Parallel to gene clustering is the important application of sample classification based on all or selected gene expressions. The gene clustering and sample classification are often undertaken separately, or in a directional manner (one as an aid for the other). However, such separation of these two tasks may occlude informative structure in the data. Here we present an algorithm for the simultaneous clustering of genes and subset selection of gene clusters for sample classification. We develop a new model selection criterion based on Rissanen's MDL (minimum description length) principle. For the first time, an MDL code length is given for both explanatory variables (genes) and response variables (sample class labels). The final output of the proposed algorithm is a sparse and interpretable classification rule based on cluster centroids or the closest genes to the centroids. RESULTS: Our algorithm for simultaneous gene clustering and subset selection for classification is applied to three publicly available data sets. For all three data sets, we obtain sparse and interpretable classification models based on centroids of clusters. At the same time, these models give competitive test error rates as the best reported methods. Compared with classification models based on single gene selections, our rules are stable in the sense that the number of clusters has a small variability and the centroids of the clusters are well correlated (or consistent) across different cross validation samples. We also discuss models where the centroids of clusters are replaced with the genes closest to the centroids. These models show comparable test error rates to models based on single gene selection, but are more sparse as well as more stable. Moreover, we comment on how the inclusion of a classification criterion affects the gene clustering, bringing out class informative structure in the data. AVAILABILITY: The methods presented in this paper have been implemented in the R language. The source code is available from the first author.

Microarray image technology is a powerful tool for monitoring the expression of thousands of genes simultaneously. Each microarray experiment produces immense amounts of image data, and efficient storage and transmission requires compression that utilizes the microarray image's structure and unique analysis goals. We have developed a progressive compression scheme for microarray images which can be either lossy or lossless. Our scheme has a coded data structure that allows fast decoding and reprocessing of image subsets, and includes summary statistics and image segmentation information. Since visual fidelity is not the end goal for microarray images, we introduce a new measure of distortion for lossy compression: the sensitivity of microarray information extraction to compression loss. We find that a lossy compression ratio of 8:1 for cDNA microarrays minimally affects downstream processing. The average lossless compression ratio is 1.83:1 for cDNA images and 2.43:1 for inkjet images, comparable to state-of-the-art lossless schemas, yet with added flexibility and information.

This paper proposes a versatile perceptual audio coding method that achieves high compression ratios and is capable of low encoding/decoding delay. It accommodates a variety of source signals (including both music and speech) with different sampling rates. It is based on separating irrelevance and redundancy reductions into independent functional units. This contrasts traditional audio coding where both are integrated within the same subband decomposition. The separation allows for the independent optimization of the irrelevance and redundancy reduction units. For both reductions, we rely on adaptive filtering and predictive coding as much as possible to minimize the delay. A psycho-acoustically controlled adaptive linear filter is used for the irrelevance reduction, and the redundancy reduction is carried out by a predictive lossless coding scheme, which is termed weighted cascaded least mean squared (WCLMS) method. Experiments are carried out on a database of moderate size which contains mono-signals of different sampling rates and varying nature (music, speech, or mixed). They show that the proposed WCLMS lossless coder outperforms other competing lossless coders in terms of compression ratios and delay, as applied to the pre-filtered signal. Moreover, a subjective listening test of the combined pre-filter/lossless coder and a state-of-the-art perceptual audio coder (PAC) shows that the new method achieves a comparable compression ratio and audio quality with a lower delay.

A novel predictive lossless coding scheme is proposed. The prediction is based on a new weighted cascaded least mean squared (WCLMS) method. To obtain both a high compression ratio and a very low encoding and decoding delay, the residuals from the prediction are encoded using either a variant of adaptive Huffman coding or a version of adaptive arithmetic coding. WCLMS is especially designed for music/speech signals. It can be used either in combination with psycho-acoustically pre-filtered signals to obtain perceptually lossless coding, or as a stand-alone lossless coder. Experiments on a database of moderate size and a variety of pre-filtered mono-signals show that the proposed lossless coder (which needs about 2 bit/sample for pre-filtered signals) outperforms competing lossless coders, such as ppmz, bzip2, Shorten, and LPAC, in terms of compression ratios. The combination of WCLMS with either of the adaptive coding schemes is also shown to achieve better compression ratios and lower delay than an earlier scheme combining WCLMS with Huffman coding over blocks of 4096 samples.

A novel predictive lossless coding scheme is proposed. The prediction is based on a new weighted cascaded least mean squared (WCLMS) method. WCLMS is especially designed for music/speech signals. It can be used either in combination with psycho-acoustically pre-filtered signals to obtain perceptually lossless coding, or as a stand-alone lossless coder. Experiments on a database of moderate size and a variety of pre-filtered mono-signals show that the proposed lossless coder (which needs about 2 bit/sample for pre-filtered signals) outperforms competing lossless coders, WaveZip, Shorten, LTAC and LPAC, in terms of compression ratios.

The method of wavelet thresholding for removing noise, or denoising, has been researched extensively due to its effectiveness and simplicity. Much of the literature has focused on developing the best uniform threshold or best basis selection. However, not much has been done to make the threshold values adaptive to the spatially changing statistics of images. Such adaptivity can improve the wavelet thresholding performance because it allows additional local information of the image (such as the identification of smooth or edge regions) to be incorporated into the algorithm. This work proposes a spatially adaptive wavelet thresholding method based on context modeling, a common technique used in image compression to adapt the coder to changing image characteristics. Each wavelet coefficient is modeled as a random variable of a generalized Gaussian distribution with an unknown parameter. Context modeling is used to estimate the parameter for each coefficient, which is then used to adapt the thresholding strategy. This spatially adaptive thresholding is extended to the overcomplete wavelet expansion, which yields better results than the orthogonal transform. Experimental results show that spatially adaptive wavelet thresholding yields significantly superior image quality and lower MSE than the best uniform thresholding with the original image assumed known.

The first part of this paper proposes an adaptive, data-driven threshold for image denoising via wavelet soft-thresholding. The threshold is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the generalized Gaussian distribution (GGD) widely used in image processing applications. The proposed threshold is simple and closed-form, and it is adaptive to each subband because it depends on data-driven estimates of the parameters. Experimental results show that the proposed method, called BayesShrink, is typically within 5% of the MSE of the best soft-thresholding benchmark with the image assumed known. It also outperforms SureShrink (Donoho and Johnstone 1994, 1995; Donoho 1995) most of the time. The second part of the paper attempts to further validate claims that lossy compression can be used for denoising. The BayesShrink threshold can aid in the parameter selection of a coder designed with the intention of denoising, and thus achieving simultaneous denoising and compression. Specifically, the zero-zone in the quantization step of compression is analogous to the threshold value in the thresholding function. The remaining coder design parameters are chosen based on a criterion derived from Rissanen's minimum description length (MDL) principle. Experiments show that this compression method does indeed remove noise significantly, especially for large noise power. However, it introduces quantization noise and should be used only if bitrate were an additional concern to denoising.

This correspondence addresses the recovery of an image from its multiple noisy copies. The standard method is to compute the weighted average of these copies. Since the wavelet thresholding technique has been shown to effectively denoise a single noisy copy, we consider in this paper combining the two operations of averaging and thresholding. Because thresholding is a nonlinear technique, averaging then thresholding or thresholding then averaging produce different estimators. By modeling the signal wavelet coefficients as Laplacian distributed and the noise as Gaussian, our investigation finds the optimal ordering to depend on the number of available copies and on the signal-to-noise ratio. We then propose thresholds that are nearly optimal under the assumed model for each ordering. With the optimal and near-optimal thresholds, the two methods yield similar performance, and both show considerable improvement over merely averaging.

We study the application of Rissanen's (1989) principle of minimum description length (MDL) to the problem of wavelet denoising and compression for natural images. After making a connection between thresholding and model selection, we derive an MDL criterion based on a Laplacian model for noiseless wavelet coefficients. We find that this approach leads to an adaptive thresholding rule. While achieving mean-squared-error performance comparable with other popular thresholding schemes, the MDL procedure tends to keep far fewer coefficients. From this property, we demonstrate that our method is an excellent tool for simultaneous denoising and compression. We make this claim precise by analyzing MDL thresholding in two optimality frameworks; one in which we measure rate and distortion based on quantized coefficients and one in which we do not quantize, but instead record rate simply as the number of nonzero coefficients.

Rissanen's (1978, 1986, 1989, 1996) minimum description length (MDL) principle is a statistical modeling principle motivated by coding theory. For exponential families we obtain pathwise expansions, to the constant order, of the predictive and mixture codelengths used in MDL. The results are useful for understanding different MDL forms.

Using in situ hyperspectral measurements collected in the Sierra Nevada Mountains in California, the authors discriminate six species of conifer trees using a recent, nonparametric statistics technique known as penalized discriminant analysis (PDA). A classification accuracy of 76% is obtained. Their emphasis is on providing an intuitive, geometric description of PDA that makes the advantages of penalization clear. PDA is a penalized version of Fisher's linear discriminant analysis (LDA) and can greatly improve upon LDA when there are a large number of highly correlated variables.

Adaptive compression methods have been a key component of many proposed subband (or wavelet) image coding techniques. This paper deals with a particular type of adaptive subband image coding where we focus on the image coder's ability to adjust itself "on the fly" to the spatially varying statistical nature of image contents. This backward adaptation is distinguished from more frequently used forward adaptation in that forward adaptation selects the best operating parameters from a predesigned set and thus uses considerable amount of side information in order for the encoder and the decoder to operate with the same parameters. Specifically, we present backward adaptive quantization using a new context-based classification technique which classifies each subband coefficient based on the surrounding quantized coefficients. We couple this classification with online parametric adaptation of the quantizer applied to each class. A simple uniform threshold quantizer is employed as the baseline quantizer for which adaptation is achieved. Our subband image coder based on the proposed adaptive classification quantization idea exhibits excellent rate-distortion performance, in particular at very low rates. For popular test images, it is comparable or superior to most of the state-of-the-art coders in the literature.

The method of wavelet thresholding for removing noise, or denoising, has been researched extensively due to its effectiveness and simplicity. Much of the work has been concentrated on finding the best uniform threshold or best basis. However, not much has been done to make this method adaptive to spatially changing statistics which is typical of a large class of images. This work proposes a spatially adaptive wavelet thresholding method based on context modeling, a common technique used in image compression to adapt the coder to the non-stationarity of images. We model each coefficient as a random variable with the generalized Gaussian prior with unknown parameters. Context modeling is used to estimate the parameters for each coefficient, which are then used to adapt the thresholding strategy. Experimental results show that spatially adaptive wavelet thresholding yields significantly superior image quality and lower MSE than optimal uniform thresholding.

This work addresses the recovery of an image from noisy observations when multiple noisy copies of the image are available. The standard method is to compute the average of these copies. Since the wavelet thresholding technique has been shown to effectively denoise a single noisy copy, it is natural to consider combining these two operations of averaging and thresholding. The first important task is to find the optimal ordering. The second issue is the threshold selection for each method. By modeling the signal wavelet coefficients as Laplacian distributed and the noise as Gaussian, our investigation finds the optimal ordering to depend on the number of available copies and on the signal-to-noise ratio. We propose thresholds that are nearly optimal under the assumed model for each ordering. With the optimal and near-optimal thresholds, the two methods yield very similar performance, and both show considerable improvement over merely averaging.

We review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling. Stochastic complexity is formulated as the solution to optimum universal coding problems extending Shannon's basic source coding theorem. The normalized maximized likelihood, mixture, and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms. We assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference. Context tree modeling, density estimation, and model selection in Gaussian linear regression serve as examples.

Some past work has proposed to use lossy compression to remove noise, based on the rationale that a reasonable compression method retains the dominant signal features more than the randomness of the noise. Building on this theme, we explain why compression (via coefficient quantization) is appropriate for filtering noise from signal by making the connection that quantization of transform coefficients approximates the operation of wavelet thresholding for denoising. That is, denoising is mainly due to the zero-zone and that the full precision of the thresholded coefficients is of secondary importance. The method of quantization is facilitated by a criterion similar to Rissanen's minimum description length principle. An important issue is the threshold value of the zero-zone (and of wavelet thresholding). For a natural image, it has been observed that its subband coefficients can be well modeled by a Laplacian distribution. With this assumption, we derive a threshold which is easy to compute and is intuitive. Experiments show that the proposed threshold performs close to optimal thresholding.

Subband image coding techniques owe much of their success to an effective use of adaptive quantization and adaptive entropy coding. It is often the case that adaptive quantization is achieved by defining a discrete set of quantizers from which one is chosen for a given set of coefficients. This type of forward adaptation thus requires that overhead information (the choice of quantizer) be sent to the decoder. Then, the quantized coefficients are transmitted using adaptive entropy coding, typically through backward adaptive arithmetic coding. We show that a combination of forward and backward adaptation methods can be used to update the quantizers thus reducing the overhead requirements while still providing good performance. Specifically, we present an algorithm where each coefficient is classified into several classes based on the past quantized data and where the quantizer to be used for each class can itself be adapted on the fly.

The article focuses on lower bound results on expected redundancy for universal coding of independent and identically distributed data on [0, 1] from parametric and nonparametric families. After reviewing existing lower bounds, we provide a new proof for minimax lower bounds on expected redundancy over nonparametric density classes. This new proof is based on the calculation of a mutual information quantity, or it utilizes the relationship between redundancy and Shannon capacity. It therefore unifies the minimax redundancy lower bound proofs in the parametric and nonparametric cases.

The problem of optimal rate universal coding is considered in the context of rate-distortion theory. A D-semifaithful universal coding scheme for discrete memoryless sources is given. The main result is a refined covering lemma based on the random coding argument and the method of types. The average codelength of the code is shown to approach its lower bound, the rate-distortion function, at a rate O(n/sup -1/log n), and this is conjectured to be optimal based on a result of A.J. Pilc (1968). Issues of constructiveness and universality are addressed.>

The results by P. Hall and E.J. Hannan (1988) on optimization of histogram density estimators with equal bin widths by minimization of the stochastic complexity are extended and sharpened in two separate ways. As the first contribution, two generalized histogram estimators are constructed. The first has unequal bin widths which, together with the number of the bins, are determined by minimization of the stochastic complexity using dynamic programming. The other estimator consists of a mixture of equal bin width estimators, each of which is defined by the associated stochastic complexity. As the main contribution in the present work, two theorems are proved, which together extend the universal coding theorems to a large class of data generating densities. The first gives an asymptotic upper bound for the code redundancy in the order of magnitude, achieved with a special predictive type of histogram estimator, which sharpens a related bound. The second theorem states that this bound cannot be improved upon by any code whatsoever.>