Multiclass probability estimation is the problem of estimating conditional probabilities of a data point belonging to a class given its covariate information. It has broad applications in statistical analysis and data science. Recently a class of weighted Support Vector Machines (wSVMs) has been developed to estimate class probabilities through ensemble learning for $K$-class problems (Wu, Zhang and Liu, 2010; Wang, Zhang and Wu, 2019), where $K$ is the number of classes. The estimators are robust and achieve high accuracy for probability estimation, but their learning is implemented through pairwise coupling, which demands polynomial time in $K$. In this paper, we propose two new learning schemes, the baseline learning and the One-vs-All (OVA) learning, to further improve wSVMs in terms of computational efficiency and estimation accuracy. In particular, the baseline learning has optimal computational complexity in the sense that it is linear in $K$. Though not being most efficient in computation, the OVA offers the best estimation accuracy among all the procedures under comparison. The resulting estimators are distribution-free and shown to be consistent. We further conduct extensive numerical experiments to demonstrate finite sample performance.
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Multiple Tensor-Times-Matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations.
In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.
We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the transport-based density representation. We then demonstrate the implications of our theory for the case of triangular Knothe-Rosenblatt (KR) transports on the $d$-dimensional unit cube, and show that both penalized and unpenalized versions of such estimators achieve minimax optimal convergence rates over H\"older classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded H\"older-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.
Real world-datasets characterized by discrete features are ubiquitous: from categorical surveys to clinical questionnaires, from unweighted networks to DNA sequences. Nevertheless, the most common unsupervised dimensional reduction methods are designed for continuous spaces, and their use for discrete spaces can lead to errors and biases. In this letter we introduce an algorithm to infer the intrinsic dimension (ID) of datasets embedded in discrete spaces. We demonstrate its accuracy on benchmark datasets, and we apply it to analyze a metagenomic dataset for species fingerprinting, finding a surprisingly small ID, of order 2. This suggests that evolutive pressure acts on a low-dimensional manifold despite the high-dimensionality of sequences' space.
We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on $n$ points. We first consider the $o(n)$-space regime and show that, when the input is a stream of all $\binom{n}{2}$ entries of the metric, for any $\alpha \ge 2$, both MST and TSP cost can be $\alpha$-approximated using $\tilde{O}(n/\alpha)$ space, and that $\Omega(n/\alpha^2)$ space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed $p$ passes over a metric stream, it still requires $\tilde{\Omega}(\sqrt{n/\alpha p^2})$ space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than $2$. We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any $\varepsilon > 0$, any one-pass $(2-\varepsilon)$-approximation of TSP cost requires $\Omega(\varepsilon^2 n^2)$ space; on the other hand, there is an $\tilde{O}(n)$ space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than $2$, when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a $(1+\varepsilon)$-approximation is achievable with $\tilde{O}(n/\varepsilon^{O(1)})$ queries. We design an algorithm that performs $\tilde{O}(n^{1.5})$ distance queries and achieves a strictly better than $2$-approximation when either the metric is known to contain a spanning tree supported on weight-$1$ edges or the algorithm is given access to a minimum spanning tree of the graph.
Many mobile manufacturers recently have adopted Dual-Pixel (DP) sensors in their flagship models for faster auto-focus and aesthetic image captures. Despite their advantages, research on their usage for 3D facial understanding has been limited due to the lack of datasets and algorithmic designs that exploit parallax in DP images. This is because the baseline of sub-aperture images is extremely narrow and parallax exists in the defocus blur region. In this paper, we introduce a DP-oriented Depth/Normal network that reconstructs the 3D facial geometry. For this purpose, we collect a DP facial data with more than 135K images for 101 persons captured with our multi-camera structured light systems. It contains the corresponding ground-truth 3D models including depth map and surface normal in metric scale. Our dataset allows the proposed matching network to be generalized for 3D facial depth/normal estimation. The proposed network consists of two novel modules: Adaptive Sampling Module and Adaptive Normal Module, which are specialized in handling the defocus blur in DP images. Finally, the proposed method achieves state-of-the-art performances over recent DP-based depth/normal estimation methods. We also demonstrate the applicability of the estimated depth/normal to face spoofing and relighting.
Propensity score weighting is widely used to improve the representativeness and correct the selection bias in the voluntary sample. The propensity score is often developed using a model for the sampling probability, which can be subject to model misspecification. In this paper, we consider an alternative approach of estimating the inverse of the propensity scores using the density ratio function satisfying the self-efficiency condition. The smoothed density ratio function is obtained by the solution to the information projection onto the space satisfying the moment conditions on the balancing scores. By including the covariates for the outcome regression models only in the density ratio model, we can achieve efficient propensity score estimation. Penalized regression is used to identify important covariates. We further extend the proposed approach to the multivariate missing case. Some limited simulation studies are presented to compare with the existing methods.
In recent years, change point detection for high dimensional data has become increasingly important in many scientific fields. Most literature develop a variety of separate methods designed for specified models (e.g. mean shift model, vector auto-regressive model, graphical model). In this paper, we provide a unified framework for structural break detection which is suitable for a large class of models. Moreover, the proposed algorithm automatically achieves consistent parameter estimates during the change point detection process, without the need for refitting the model. Specifically, we introduce a three-step procedure. The first step utilizes the block segmentation strategy combined with a fused lasso based estimation criterion, leads to significant computational gains without compromising the statistical accuracy in identifying the number and location of the structural breaks. This procedure is further coupled with hard-thresholding and exhaustive search steps to consistently estimate the number and location of the break points. The strong guarantees are proved on both the number of estimated change points and the rates of convergence of their locations. The consistent estimates of model parameters are also provided. The numerical studies provide further support of the theory and validate its competitive performance for a wide range of models. The developed algorithm is implemented in the R package LinearDetect.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
In structure learning, the output is generally a structure that is used as supervision information to achieve good performance. Considering the interpretation of deep learning models has raised extended attention these years, it will be beneficial if we can learn an interpretable structure from deep learning models. In this paper, we focus on Recurrent Neural Networks (RNNs) whose inner mechanism is still not clearly understood. We find that Finite State Automaton (FSA) that processes sequential data has more interpretable inner mechanism and can be learned from RNNs as the interpretable structure. We propose two methods to learn FSA from RNN based on two different clustering methods. We first give the graphical illustration of FSA for human beings to follow, which shows the interpretability. From the FSA's point of view, we then analyze how the performance of RNNs are affected by the number of gates, as well as the semantic meaning behind the transition of numerical hidden states. Our results suggest that RNNs with simple gated structure such as Minimal Gated Unit (MGU) is more desirable and the transitions in FSA leading to specific classification result are associated with corresponding words which are understandable by human beings.
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