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In the statistical literature, sparse modeling is the standard approach to achieve improvements in prediction tasks and interpretability. Alternatively, in the seminal paper "Statistical Modeling: The Two Cultures," Breiman (2001) advocated for the adoption of algorithmic approaches to generate ensembles to achieve superior prediction accuracy than single-model methods at the cost of loss of interpretability. In a recent important and critical paper, Rudin (2019) argued that blackbox algorithmic approaches should be avoided for high-stakes decisions and that the tradeoff between accuracy and interpretability is a myth. In response to this recent change in philosophy, we generalize best subset selection (BSS) to best split selection (BSpS), a data-driven approach aimed at finding the optimal split of predictor variables among the models of an ensemble. The proposed methodology results in an ensemble of sparse and diverse models that provide possible mechanisms that explain the relationship between the predictors and the response. The high computational cost of BSpS motivates the need for computational tractable ways to approximate the exhaustive search, and we benchmark one such recent proposal by Christidis et al. (2020) based on a multi-convex relaxation. Our objective with this article is to motivate research in this new exciting field with great potential for data analysis tasks for high-dimensional data.

This article is motivated by studying multisensory effects on brain activities in intracranial electroencephalography (iEEG) experiments. Differential brain activities to multisensory stimulus presentations are zero in most regions and non-zero in some local regions, yielding locally sparse functions. Such studies are essentially a function-on-scalar regression problem, with interest being focused not only on estimating nonparametric functions but also on recovering the function supports. We propose a weighted group bridge approach for simultaneous function estimation and support recovery in function-on-scalar mixed effect models, while accounting for heterogeneity present in functional data. We use B-splines to transform sparsity of functions to its sparse vector counterpart of increasing dimension, and propose a fast non-convex optimization algorithm using nested alternative direction method of multipliers (ADMM) for estimation. Large sample properties are established. In particular, we show that the estimated coefficient functions are rate optimal in the minimax sense under the $L_2$ norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the $L_{\infty}$ norm that leads to a sparsistency property under $\delta$-sparsity, and provide a simple sufficient regularity condition under which a strict sparsistency property is established. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulation and an application to a novel iEEG dataset to study multisensory integration. We integrate the proposed method into RAVE, an R package that gains increasing popularity in the iEEG community.

It is well-known that an algorithm exists which approximates the NP-complete problem of Set Cover within a factor of ln(n), and it was recently proven that this approximation ratio is optimal unless P = NP. This optimality result is the product of many advances in characterizations of NP, in terms of interactive proof systems and probabilistically checkable proofs (PCP), and improvements to the analyses thereof. However, as a result, it is difficult to extract the development of Set Cover approximation bounds from the greater scope of proof system analysis. This paper attempts to present a chronological progression of results on lower-bounding the approximation ratio of Set Cover. We analyze a series of proofs of progressively better bounds and unify the results under similar terminologies and frameworks to provide an accurate comparison of proof techniques and their results. We also treat many preliminary results as black-boxes to better focus our analysis on the core reductions to Set Cover instances. The result is alternative versions of several hardness proofs, beginning with initial inapproximability results and culminating in a version of the proof that ln(n) is a tight lower bound.

Often machine learning and statistical models will attempt to describe the majority of the data. However, there may be situations where only a fraction of the data can be fit well by a linear regression model. Here, we are interested in a case where such inliers can be identified by a Disjunctive Normal Form (DNF) formula. We give a polynomial time algorithm for the conditional linear regression task, which identifies a DNF condition together with the linear predictor on the corresponding portion of the data. In this work, we improve on previous algorithms by removing a requirement that the covariances of the data satisfying each of the terms of the condition have to all be very similar in spectral norm to the covariance of the overall condition.

In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.

Many well-known matrices $Z$ are associated to fast transforms corresponding to factorizations of the form $Z = X^J \ldots X^1$, where each factor $X^\ell$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations. Our first main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits a unique factorization into $J$ butterfly factors (where $N = 2^J$), and that the factors can be recovered by a hierarchical factorization method. This contrasts with existing approaches which fit the product of the butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorizations of the Hadamard or the Discrete Fourier Transform matrices of size $2^J$. Computing such factorizations costs $\mathcal{O}(N^2)$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting that was recently established. While the butterfly structure corresponds to a fixed prescribed support for each factor, our second contribution is to obtain identifiability results with more general families of allowed sparsity patterns, taking into account permutation ambiguities when they are unavoidable. Typically, we show through the hierarchical paradigm that the butterfly factorization of the Discrete Fourier Transform matrix of size $2^J$ admits a unique sparse factorization into $J$ factors, when enforcing only $2$-sparsity by column and a block-diagonal structure on each factor.

We study sparse linear regression over a network of agents, modeled as an undirected graph (with no centralized node). The estimation problem is formulated as the minimization of the sum of the local LASSO loss functions plus a quadratic penalty of the consensus constraint -- the latter being instrumental to obtain distributed solution methods. While penalty-based consensus methods have been extensively studied in the optimization literature, their statistical and computational guarantees in the high dimensional setting remain unclear. This work provides an answer to this open problem. Our contribution is two-fold. First, we establish statistical consistency of the estimator: under a suitable choice of the penalty parameter, the optimal solution of the penalized problem achieves near optimal minimax rate $\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value, $d$ is the ambient dimension, and $N$ is the total sample size in the network -- this matches centralized sample rates. Second, we show that the proximal-gradient algorithm applied to the penalized problem, which naturally leads to distributed implementations, converges linearly up to a tolerance of the order of the centralized statistical error -- the rate scales as $\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical results demonstrate the tightness of the derived sample rate and convergence rate scalings.

Systems with stochastic time delay between the input and output present a number of unique challenges. Time domain noise leads to irregular alignments, obfuscates relationships and attenuates inferred coefficients. To handle these challenges, we introduce a maximum likelihood regression model that regards stochastic time delay as an "error" in the time domain. For a certain subset of problems, by modelling both prediction and time errors it is possible to outperform traditional models. Through a simulated experiment of a univariate problem, we demonstrate results that significantly improve upon Ordinary Least Squares (OLS) regression.

Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm for this task that greatly outperforms existing methods. Experiments using hundreds of matrices from diverse domains show that it often runs $100\times$ faster than exact matrix products and $10\times$ faster than current approximate methods. In the common case that one matrix is known ahead of time, our method also has the interesting property that it requires zero multiply-adds. These results suggest that a mixture of hashing, averaging, and byte shuffling$-$the core operations of our method$-$could be a more promising building block for machine learning than the sparsified, factorized, and/or scalar quantized matrix products that have recently been the focus of substantial research and hardware investment.

Non-maximum suppression (NMS) is essential for state-of-the-art object detectors to localize object from a set of candidate locations. However, accurate candidate location sometimes is not associated with a high classification score, which leads to object localization failure during NMS. In this paper, we introduce a novel bounding box regression loss for learning bounding box transformation and localization variance together. The resulting localization variance exhibits a strong connection to localization accuracy, which is then utilized in our new non-maximum suppression method to improve localization accuracy for object detection. On MS-COCO, we boost the AP of VGG-16 faster R-CNN from 23.6% to 29.1% with a single model and nearly no additional computational overhead. More importantly, our method is able to improve the AP of ResNet-50 FPN fast R-CNN from 36.8% to 37.8%, which achieves state-of-the-art bounding box refinement result.