The trade-off between accuracy and interpretability has long been a challenge in machine learning (ML). This tension is particularly significant for emerging *interpretable-by-design* methods, which aim to redesign ML algorithms for trustworthy interpretability but often sacrifice accuracy in the process. In this paper, we address this gap by investigating the impact of deviations in concept representations—an essential component of interpretable models—on prediction performance and propose a novel framework to mitigate these effects. The framework builds on the principle of optimizing concept embeddings under constraints that preserve interpretability. Using a generative model as a test-bed, we rigorously prove that our algorithm achieves zero loss while progressively enhancing the interpretability of the resulting model. Additionally, we evaluate the practical performance of our proposed framework in generating explainable predictions for image classification tasks across various benchmarks. Compared to existing explainable methods, our approach not only improves prediction accuracy while preserving model interpretability across various large-scale benchmarks but also achieves this with significantly lower computational cost.

We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $X^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $UU^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $X^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r’\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r’\geq r$, we show that, with $\widetilde\Omega(dr^9)$ observations, GD with an initial point $\|U_0\| \leq O(\epsilon)$ converges near-linearly to an $\epsilon$-neighborhood of $X^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r’$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r’=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $\epsilon>0$, provided the initial point satisfies $\|U_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis.

In this work, we study the problem of common and unique feature extraction from noisy data. When we have $N$ observation matrices from $N$ different and associated sources corrupted by sparse and potentially gross noise, can we recover the common and unique components from these noisy observations? This is a challenging task as the number of parameters to estimate is approximately thrice the number of observations. Despite the difficulty, we propose an intuitive alternating minimization algorithm called triple component matrix factorization (TCMF) to recover the three components exactly. TCMF is distinguished from existing works in literature thanks to two salient features. First, TCMF is a principled method to separate the three components given noisy observations provably. Second, the bulk of the computation in TCMF can be distributed. On the technical side, we formulate the problem as a constrained nonconvex nonsmooth optimization problem. Despite the intricate nature of the problem, we provide a Taylor series characterization of its solution by solving the corresponding Karush–Kuhn–Tucker conditions. Using this characterization, we can show that the alternating minimization algorithm makes significant progress at each iteration and converges into the ground truth at a linear rate. Numerical experiments in video segmentation and anomaly detection highlight the superior feature extraction abilities of TCMF.

We introduce a relevant yet challenging problem named Personalized Dictionary Learning (PerDL), where the goal is to learn sparse linear representations from heterogeneous datasets that share some commonality. In PerDL, we model each dataset's shared and unique features as global and local dictionaries. Challenges for PerDL not only are inherited from classical dictionary learning(DL), but also arise due to the unknown nature of the shared and unique features. In this paper, we rigorously formulate this problem and provide conditions under which the global and local dictionaries can be provably disentangled. Under these conditions, we provide a meta-algorithm called Personalized Matching and Averaging (PerMA) that can recover both global and local dictionaries from heterogeneous datasets. PerMA is highly efficient; it converges to the ground truth at a linear rate under suitable conditions. Moreover, it automatically borrows strength from strong learners to improve the prediction of weak learners. As a general framework for extracting global and local dictionaries, we show the application of PerDL in different learning tasks, such as training with imbalanced datasets and video surveillance.

In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with $\ell_1$-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the rank of the true solution is unknown and over-estimated instead. The over-estimation of the rank gives rise to an over-parameterized model in which there are more degrees of freedom than needed. Such over-parameterization may lead to overfitting, or adversely affect the performance of the algorithm. We prove that a simple SubGM with small initialization is agnostic to both over-parameterization and noise in the measurements. In particular, we show that small initialization nullifies the effect of over-parameterization on the performance of SubGM, leading to an exponential improvement in its convergence rate. Moreover, we provide the first unifying framework for analyzing the behavior of SubGM under both outlier and Gaussian noise models, showing that SubGM converges to the true solution, even under arbitrarily large and arbitrarily dense noise values, and, perhaps surprisingly, even if the globally optimal solutions do not correspond to the ground truth. At the core of our results is a robust variant of restricted isometry property, called Sign-RIP, which controls the deviation of the sub-differential of the $\ell_1$-loss from that of an ideal, expected loss. As a byproduct of our results, we consider a subclass of robust low-rank matrix recovery with Gaussian measurements, and show that the number of required samples to guarantee the global convergence of SubGM is independent of the over-parameterized rank.

We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally rank deficient, then its rank deficiency certifies it as being globally optimal. This way of certifying global optimality necessarily requires the search rank $r$ of the current iterate $X$ to be overparameterized with respect to the rank $r^{\star}$ of the global minimizer $X^{\star}$. Unfortunately, overparameterization significantly slows down the convergence of gradient descent, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$, even when $\phi$ is strongly convex. In this paper, we propose an inexpensive preconditioner that restores the convergence rate of gradient descent back to linear in the overparameterized case, while also making it agnostic to possible ill-conditioning in the global minimizer $X^{\star}$.

In this paper, we study the problem of inferring spatially-varying Gaussian Markov random fields (SV-GMRF) where the goal is to learn a network of sparse, context-specific GMRFs representing network relationships between genes. An important application of SV-GMRFs is in inference of gene regulatory networks from spatially-resolved transcriptomics datasets. The current work on inference of SV-GMRFs are based on the regularized maximum likelihood estimation (MLE) and suffer from overwhelmingly high computational cost due to their highly nonlinear nature. To alleviate this challenge, we propose a simple and efficient optimization problem in lieu of MLE that comes equipped with strong statistical and computational guarantees. Our proposed optimization problem is extremely efficient in practice: we can solve instances of SV-GMRFs with more than 2 million variables in less than 2 minutes. We apply the developed framework to study how gene regulatory networks in Glioblastoma are spatially rewired within tissue, and identify prominent activity of the transcription factor HES4 and ribosomal proteins as characterizing the gene expression network in the tumor peri-vascular niche that is known to harbor treatment resistant stem cells.

This work characterizes the effect of depth on the optimization landscape of linear regression, showing that, despite their nonconvexity, deeper models have more desirable optimization landscape. We consider a robust and over-parameterized setting, where a subset of measurements are grossly corrupted with noise, and the true linear model is captured via an $N$-layer diagonal linear neural network. On the negative side, we show that this problem does not have a benign landscape: given any $N\geq 1$, with constant probability, there exists a solution corresponding to the ground truth that is neither local nor global minimum. However, on the positive side, we prove that, for any $N$-layer model with $N\geq 2$, a simple sub-gradient method becomes oblivious to such “problematic” solutions; instead, it converges to a balanced solution that is not only close to the ground truth but also enjoys a flat local landscape, thereby eschewing the need for “early stopping”. Lastly, we empirically verify that the desirable optimization landscape of deeper models extends to other robust learning tasks, including deep matrix recovery and deep ReLU networks with $\ell_1$-loss.

We study the problem of inferring time-varying Gaussian Markov random fields, where the underlying graphical model is both sparse and changes {sparsely} over time. Most of the existing methods for the inference of time-varying Markov random fields (MRFs) rely on the \textit{regularized maximum likelihood estimation} (MLE), that typically suffer from weak statistical guarantees and high computational time. Instead, we introduce a new class of constrained optimization problems for the inference of sparsely-changing Gaussian MRFs (GMRFs). The proposed optimization problem is formulated based on the exact $\ell_0$ regularization, and can be solved in near-linear time and memory. Moreover, we show that the proposed estimator enjoys a provably small estimation error. We derive sharp statistical guarantees in the high-dimensional regime, showing that such problems can be learned with as few as one sample per time period. Our proposed method is extremely efficient in practice: it can accurately estimate sparsely-changing GMRFs with more than 500 million variables in less than one hour.

In practical instances of nonconvex matrix factorization, the rank of the true solution $r^{\star}$ is often unknown, so the rank $r$of the model can be over-specified as $r>r^{\star}$. This over-parameterized regime of matrix factorization significantly slows down the convergence of local search algorithms, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$. We propose an inexpensive preconditioner for the matrix sensing variant of nonconvex matrix factorization that restores the convergence rate of gradient descent back to linear, even in the over-parameterized case, while also making it agnostic to possible ill-conditioning in the ground truth. Classical gradient descent in a neighborhood of the solution slows down due to the need for the model matrix factor to become singular. Our key result is that this singularity can be corrected by $\ell_{2}$ regularization with a specific range of values for the damping parameter. In fact, a good damping parameter can be inexpensively estimated from the current iterate. The resulting algorithm, which we call preconditioned gradient descent or PrecGD, is stable under noise, and converges linearly to an information theoretically optimal error bound. Our numerical experiments find that PrecGD works equally well in restoring the linear convergence of other variants of nonconvex matrix factorization in the over-parameterized regime.

This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the RPCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the RPCA as a non-convex and non-smooth $\ell_1$ optimization problem with a significantly smaller number of variables. In this work, we show that the low-dimensional formulation of the symmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiro approach has benign landscape, i.e., 1) it does not have any spurious local solution, 2) has a unique global solution, and 3) its unique global solution coincides with the true components. An implication of this result is that simple local search algorithms are guaranteed to achieve a zero global optimality gap when directly applied to the low-dimensional formulation. Furthermore, we provide strong deterministic and probabilistic guarantees for the exact recovery of the true principal components. In particular, it is shown that a constant fraction of the measurements could be grossly corrupted and yet they would not create any spurious local solution.

Graphical Lasso (GL) is a popular method for learning the structure of an undirected graphical model, which is based on an $l_1$ regularization technique. The objective of this paper is to compare the computationally-heavy GL technique with a numerically-cheap heuristic method that is based on simply thresholding the sample covariance matrix. To this end, two notions of sign-consistent and inverse-consistent matrices are developed, and then it is shown that the thresholding and GL methods are equivalent if: (i) the thresholded sample covariance matrix is both sign-consistent and inverse-consistent, and (ii) the gap between the largest thresholded and the smallest un-thresholded entries of the sample covariance matrix is not too small. By building upon this result, it is proved that the GL method---as a conic optimization problem---has an explicit closed-form solution if the thresholded sample covariance matrix has an acyclic structure. This result is then generalized to arbitrary sparse support graphs, where a formula is found to obtain an approximate solution of GL. Furthermore, it is shown that the approximation error of the derived explicit formula decreases exponentially fast with respect to the length of the minimum-length cycle of the sparsity graph. The developed results are demonstrated on synthetic data, functional MRI data, traffic flows for transportation networks, and massive randomly generated data sets. We show that the proposed method can obtain an accurate approximation of the GL for instances with the sizes as large as $80,000\times 80,000$ (more than 3.2 billion variables) in less than 30 minutes on a standard laptop computer running MATLAB, while other state-of-the-art methods do not converge within 4 hours

The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as “graphical lasso”, but its computational cost becomes prohibitive for large data sets. A recently line of results showed{–}under mild assumptions{–}that the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix and solving a maximum determinant matrix completion (MDMC) problem. This paper proves an extension of this result, and describes a Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $\epsilon$-accurate solution in $O(n\log(1/\epsilon))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve the associated MDMC problems with as many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.