We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees for the maximum likelihood estimator significantly improve upon the currently known results. A variety of techniques are utilized and innovated upon, including Chernoff-type inequalities and empirical Bernstein bounds. We illustrate our results in synthetic and real-world experiments. Finally, we apply our proposed framework to a basic selective inference problem, where we estimate the most frequent probabilities in a sample.

Estimating the entropy of a discrete random variable is a fundamental problem in information theory and related fields. This problem has many applications in various domains, including machine learning, statistics, and data compression. Over the years, a variety of estimation schemes have been suggested. However, despite significant progress, most methods still struggle when the sample is small, compared to the variable's alphabet size. In this work, we introduce a practical solution to this problem, which extends the work of McAllester and Statos. The proposed scheme uses the generalization abilities of cross-entropy estimation in deep neural networks (DNNs) to introduce improved entropy estimation accuracy. Furthermore, we introduce a family of estimators for related information-theoretic measures, such as conditional entropy and mutual information (MI). We show that these estimators are strongly consistent and demonstrate their performance in a variety of use cases. First, we consider large alphabet entropy estimation. Then, we extend the scope to MI estimation. Next, we apply the proposed scheme to conditional MI estimation, as we focus on independence testing tasks. Finally, we study a transfer entropy (TE) estimation problem. The proposed estimators demonstrate improved performance compared to existing methods in all of these setups.

Consider a finite sample from an unknown distribution over a countable alphabet. The missing mass refers to the probability of symbols that do not appear in the sample. Missing mass estimation is a fundamental problem in statistics, information theory and related fields, which dates back to the early work of Laplace, and the more recent seminal contribution of Good and Turing. Most popular missing mass estimation schemes are universal, in the sense that they preform well for every possible distribution. Interestingly, the worst-case distribution, for which these schemes perform the worst, is known to be uniform. On the other hand, real-world distributions are typically heavy-tailed. This means that current frameworks may be over-pessimistic, in many cases of interest. In this work we suggest a data-dependent estimation scheme to address this caveat. Specifically, we infer a subset of distributions from the sample, and control the worst-case performance only over that subset. Our suggested scheme demonstrates improved performance guarantees compared to alternative methods.

Consider a finite sample from an unknown distribution over a countable alphabet. The occupancy probability (OP) refers to the total probability of symbols that appear exactly k times in the sample. Estimating the OP is a basic problem in large alphabet modeling, with a variety of applications in machine learning, statistics and information theory. The Good-Turing (GT) framework is perhaps the most popular OP estimation scheme. Classical results show that the GT estimator converges to the OP, for every k independently. In this work we introduce new exact convergence guarantees for the GT estimator, based on worst-case mean squared error analysis. Our scheme improves upon currently known results. Further, we introduce a novel simultaneous convergence rate, for any desired set of occupancy probabilities. This allows us to quantify the unified performance of OP estimators, and introduce a novel estimation framework with favorable convergence guarantees.

The Good-Turing (GT) estimator is perhaps the most popular framework for modelling large alphabet distributions. Classical results show that the GT estimator convergences to the occupancy probability, formally defined as the total probability of words that appear exactly k times in the sample. In this work we introduce new convergence guarantees for the GT estimator, based on worst-case MSE analysis. Our results refine and improve upon currently known bounds. Importantly, we introduce a simultaneous convergence rate to the entire collection of occupancy probabilities.

Typically, real-world stochastic processes are not easy to analyze. In this paper, we study the representation of different stochastic process as a memoryless innovation process triggering a dynamic system. We show that such a representation is always feasible for innovation processes taking values over a continuous set. However, the problem becomes more challenging when the alphabet size of the innovation is finite. In this case, we introduce both lossless and lossy frameworks, and provide closed-form solutions and practical algorithmic methods. In addition, we discuss the properties and uniqueness of our suggested approach. Finally, we show that the innovation representation problem has many applications. We focus our attention on entropic causal inference, which has recently demonstrated promising performance, compared to alternative methods.

A loss function measures the discrepancy between the true values and their estimated fits, for a given instance of data. In classification problems, a loss function is said to be proper if a minimizer of the expected loss is the true underlying probability. We show that for binary classification, the divergence associated with smooth, proper, and convex loss functions is upper bounded by the Kullback-Leibler (KL) divergence, to within a normalization constant. This implies that by minimizing the logarithmic loss associated with the KL divergence, we minimize an upper bound to any choice of loss from this set. As such the logarithmic loss is universal in the sense of providing performance guarantees with respect to a broad class of accuracy measures. Importantly, this notion of universality is not problem-specific, enabling its use in diverse applications, including predictive modeling, data clustering and sample complexity analysis. Generalizations to arbitary finite alphabets are also developed. The derived inequalities extend several well-known $f$ -divergence results.

A loss function measures the discrepancy between the true values (observations) and their estimated fits, for a given instance of data. A loss function is said to be proper (unbiased, Fisher consistent) if the fits are defined over a unit simplex, and the minimizer of the expected loss is the true underlying probability of the data. Typical examples are the zero-one loss, the quadratic loss and the Bernoulli log-likelihood loss (log-loss). In this work we show that for binary classification problems, the divergence associated with smooth, proper and convex loss functions is bounded from above by the Kullback-Leibler (KL) divergence, up to a multiplicative normalization constant. It implies that by minimizing the log-loss (associated with the KL divergence), we minimize an upper bound to any choice of loss functions from this set. This property justifies the broad use of log-loss in regression, decision trees, deep neural networks and many other applications. In addition, we show that the KL divergence bounds from above any separable Bregman divergence that is convex in its second argument (up to a multiplicative normalization constant). This result introduces a new set of divergence inequalities, similar to the well-known Pinsker inequality.

Recursive partitioning methods producing tree-like models are a long standing staple of predictive modeling. However, a fundamental flaw in the partitioning (or splitting) rule of commonly used tree building methods precludes them from treating different types of variables equally. This most clearly manifests in these methods' inability to properly utilize categorical variables with a large number of categories, which are ubiquitous in the new age of big data. We propose a framework to splitting using leave-one-out (LOO) cross validation (CV) for selecting the splitting variable, then performing a regular split (in our case, following CART's approach) for the selected variable. The most important consequence of our approach is that categorical variables with many categories can be safely used in tree building and are only chosen if they contribute to predictive power. We demonstrate in extensive simulation and real data analysis that our splitting approach significantly improves the performance of both single tree models and ensemble methods that utilize trees. Importantly, we design an algorithm for LOO splitting variable selection which under reasonable assumptions does not substantially increase the overall computational complexity compared to CART for two-class classification.

Large alphabet source coding is a basic and well-studied problem in data compression. It has many applications, such as compression of natural language text, speech, and images. The classic perception of most commonly used methods is that a source is best described over an alphabet, which is at least as large as the observed alphabet. In this paper, we challenge this approach and introduce a conceptual framework in which a large alphabet source is decomposed into “as statistically independent as possible” components. This decomposition allows us to apply entropy encoding to each component separately, while benefiting from their reduced alphabet size. We show that in many cases, such decomposition results in a sum of marginal entropies which is only slightly greater than the entropy of the source. Our suggested algorithm, based on a generalization of the binary independent component analysis, is applicable for a variety of large alphabet source coding setups. This includes the classical lossless compression, universal compression, and high-dimensional vector quantization. In each of these setups, our suggested approach outperforms most commonly used methods. Moreover, our proposed framework is significantly easier to implement in most of these cases.

Encoding a sequence of independent symbols over a large alphabet size is a challenging problem with applications in many fields. The most widely used adaptive entropy coding techniques (namely, arithmetic and Huffman coding) are known to achieve an average codeword length which may be significantly greater than the empirical entropy of the sequence, as the alphabet size increases. In this work we introduce an efficient and easy-to-implement method for large alphabet adaptive encoding. We propose a conceptual framework in which a sequence of symbols, over a large alphabet size, is decomposed into multiple "almost independent" sequences over a smaller alphabet. Then each of these sequences is encoded separately. This way, we allow encoding of small alphabet sequences, at the cost of the "remaining dependence" among the sequences. We demonstrate the advantages of our suggested scheme through a series of theorems and experiments, showing it reduces both the average codeword length and the compression runtime in many large alphabet setups.

Ensemble methods are considered among the state-of-the-art predictive modeling approaches. Applied to modern big data, these methods often require a large number of sub-learners, where the complexity of each learner typically grows with the size of the dataset. This phenomenon results in an increasing demand for storage space, which may be very costly. This problem mostly manifests in a subscriber based environment, where a user-specific ensemble needs to be stored on a personal device with strict storage limitations (such as a cellular device). In this work we introduce a novel method for lossless compression of tree-based ensemble methods, focusing on Random Forests. Our suggested method is based on probabilistic modeling of the ensemble's trees, followed by model clustering via Bregman divergence. This allows us to find a minimal set of models that provides an accurate description of the trees, and at the same time is small enough to store and maintain. Our compression scheme demonstrates high compression rates on a variety of modern datasets. Importantly, our scheme enables predictions from the compressed format and a perfect reconstruction of the original ensemble.

In this paper we present an algorithmic approach for fitting isotonic models under convex, yet non-differentiable, loss functions. It is a generalization of the greedy non-regret approach proposed by Luss and Rosset (2014) for differentiable loss functions, taking into account the sub-gradiental extensions required. We prove that our suggested algorithm solves the isotonic modeling problem while maintaining favorable computational and statistical properties. As our suggested algorithm may be used for any non-differentiable loss function, we focus our interest on isotonic modeling for either regression or two-class classification with appropriate log-likelihood loss and lasso penalty on the fitted values. This combination allows us to maintain the non-parametric nature of isotonic modeling, while controlling model complexity through regularization. We demonstrate the efficiency and usefulness of this approach on both synthetic and real world data. An implementation of our suggested solution is publicly available from the first author's website (https://sites.google.com/site/amichaipainsky/software).

Independent component analysis (ICA) is a statistical method for transforming an observable multi-dimensional random vector into components that are as statistically independent as possible from each other. Usually, the ICA framework assumes a model according to which the observations are generated (such as a linear transformation with additive noise). ICA over finite fields is a special case of ICA in which both the observations and the independent components are over a finite alphabet. In this paper, we consider a generalization of this framework in which an observation vector is decomposed to its independent components (as much as possible) with no prior assumption on the way it was generated. This generalization is also known as Barlow's minimal redundancy representation problem and is considered an open problem. We propose several theorems and show that this hard problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. Our contribution provides the first efficient set of solutions to Barlow's problem. The minimal redundancy representation (also known as factorial code) has many applications, mainly in the fields of neural networks and deep learning. The binary ICA is also shown to have applications in several domains, including medical diagnosis, multi-cluster assignment, network tomography, and internet resource management. In this paper, we show that this formulation further applies to multiple disciplines in source coding, such as predictive coding, distributed source coding, and coding of large alphabet sources.

Many applications of universal compression involve sources such as text, speech and image, whose alphabet is extremely large. In this work we propose a conceptual framework in which a large alphabet memory less source is decomposed into multiple 'as independent as possible' sources whose alphabet is much smaller. This way we slightly increase the average codeword length as the compressed symbols are no longer perfectly independent, but at the same time significantly reduce the overhead redundancy resulted by the large alphabet of the observed source. Our proposed algorithm, based on a generalization of the Binary Independent Component Analysis, shows to efficiently find the ideal trade-off so that the overall compression size is minimal. We demonstrate our framework on memory less draws from a variety of natural languages and show that the redundancy we achieve is remarkably smaller than most commonly used methods.

Independent component analysis (ICA) is a statistical method for transforming an observed multidimensional random vector into components that are as statistically independent as possible from each other. Usually the ICA framework assumes a model according to which the observations are generated (generative function, additive noise). Binary ICA (BICA) is a special case of ICA in which both the observations and the independent components are over the binary field GF(2). In this work we introduce a generalized BICA framework in which an observation vector is decomposed to its independent components (as much as possible) with no prior assumption on the way it was generated. We propose several theorems and show that this NP hard problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear programs. BICA was shown to have applications in many domains including medical diagnosis, multi-cluster assignment, network tomography and internet resource management. We suggest that BICA also applies in source coding; we argue that instead of generating statistically independent prediction errors, as in predictive coding, an improved encoder shall assemble a vector of observations and apply the generalized BICA on it. This is shown to achieve improved performance at the cost of introducing some time delay (working in batch).

Memoryless processes hold many theoretical and practical advantages. They are easy to describe, analyze, store and encrypt. They can also be seen as the essence of a family of regression processes, or as an innovation process triggering a dynamic system. The Gram-Schmidt procedure suggests a linear sequential method of whitening (decorrelating) any stochastic process. Applied on a Gaussian process, memorylessness (that is, statistical independence) is guaranteed. It is not clear however, how to sequentially construct a memoryless process from a non-Gaussian process. In this paper we present a non-linear sequential method to generate a memoryless process from any given Markov process under varying objectives and constraints. We differentiate between lossless and lossy methods, closed form and algorithmic solutions and discuss the properties and uniqueness of our suggested methods.

The availability of large microarray data has led to a growing interest in biclustering methods in the past decade. Several algorithms have been proposed to identify subsets of genes and conditions according to different similarity measures and under varying constraints. In this paper we focus on the exclusive row biclusteing problem (also known as projected clustering) for gene expression data sets, in which each row can only be a member of a single bicluster while columns can participate in multiple clusters. This type of biclustering may be adequate, for example, for clustering groups of cancer patients where each patient (row) is expected to be carrying only a single type of cancer, while each cancer type is associated with multiple (and possibly overlapping) genes (columns). In this paper we present a novel method to identify these exclusive row biclusters through a combination of existing biclustering algorithms and combinatorial auction techniques. We devise an approach for tuning the threshold for our algorithm based on comparison to a null model in the spirit of the Gap statistic approach. We demonstrate our approach on both synthetic and real-world gene expression data and show its power in identifying large span non-overlapping rows sub matrices, while considering their unique nature. The Gap statistic approach succeeds in identifying appropriate thresholds in all our examples.