Neural Information Processing Systems http://nips.cc/

We study robust subspace estimation in the streaming and distributed settings.

In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best approximates the input points. More precisely, for a given $p\geq 1$, we aim to minimize the $p$th power of the $\ell_p$ norm of the error vector $(\|a_1-\bm{P}a_1\|,\ldots,\|a_n-\bm{P}a_n\|)$, where $\bm{P}$ denotes the projection matrix onto the subspace and the norms are Euclidean. In \emph{constrained} subspace approximation (CSA), we additionally have constraints on the projection matrix $\bm{P}$. In its most general form, we require $\bm{P}$ to belong to a given subset $\mathcal{S}$ that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either $(1+\varepsilon)$-multiplicative or $\varepsilon$-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to {\it fair} subspace approximation, $k$-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for $k$-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.

The $\ell_p$ subspace approximation problem is an NP-hard low rank approximation problem that generalizes the median hyperplane problem ($p = 1$), principal component analysis ($p = 2$), and the center hyperplane problem ($p = \infty$). A popular approach to cope with the NP-hardness of this problem is to compute a strong coreset, which is a small weighted subset of the input points which simultaneously approximates the cost of every $k$-dimensional subspace, typically to $(1+\varepsilon)$ relative error for a small constant $\varepsilon$. We obtain the first algorithm for constructing a strong coreset for $\ell_p$ subspace approximation with a nearly optimal dependence on the rank parameter $k$, obtaining a nearly linear bound of $\tilde O(k)\mathrm{poly}(\varepsilon^{-1})$ for $p<2$ and $\tilde O(k^{p/2})\mathrm{poly}(\varepsilon^{-1})$ for $p>2$. Prior constructions either achieved a similar size bound but produced a coreset with a modification of the original points [SW18, FKW21], or produced a coreset of the original points but lost $\mathrm{poly}(k)$ factors in the coreset size [HV20, WY23]. Our techniques also lead to the first nearly optimal online strong coresets for $\ell_p$ subspace approximation with similar bounds as the offline setting, resolving a problem of [WY23]. All prior approaches lose $\mathrm{poly}(k)$ factors in this setting, even when allowed to modify the original points.

Large models and enormous data are essential driving forces of the unprecedented successes achieved by modern algorithms, especially in scientific computing and machine learning. Nevertheless, the growing dimensionality and model complexity, as well as the non-negligible workload of data pre-processing, also bring formidable costs to such successes in both computation and data aggregation. As the deceleration of Moore's Law slackens the cost reduction of computation from the hardware level, fast heuristics for expensive classical routines and efficient algorithms for exploiting limited data are increasingly indispensable for pushing the limit of algorithm potency. This thesis explores some of such algorithms for fast execution and efficient data utilization. From the computational efficiency perspective, we design and analyze fast randomized low-rank decomposition algorithms for large matrices based on "matrix sketching", which can be regarded as a dimension reduction strategy in the data space. These include the randomized pivoting-based interpolative and CUR decomposition discussed in Chapter 2 and the randomized subspace approximations discussed in Chapter 3. From the sample efficiency perspective, we focus on learning algorithms with various incorporations of data augmentation that improve generalization and distributional robustness provably. Specifically, Chapter 4 presents a sample complexity analysis for data augmentation consistency regularization where we view sample efficiency from the lens of dimension reduction in the function space. Then in Chapter 5, we introduce an adaptively weighted data augmentation consistency regularization algorithm for distributionally robust optimization with applications in medical image segmentation.

We consider the problem of subset selection for $\ell_{p}$ subspace approximation, that is, to efficiently find a \emph{small} subset of data points such that solving the problem optimally for this subset gives a good approximation to solving the problem optimally for the original input. Previously known subset selection algorithms based on volume sampling and adaptive sampling \cite{DeshpandeV07}, for the general case of $p \in [1, \infty)$, require multiple passes over the data. In this paper, we give a one-pass subset selection with an additive approximation guarantee for $\ell_{p}$ subspace approximation, for any $p \in [1, \infty)$. Earlier subset selection algorithms that give a one-pass multiplicative $(1+\epsilon)$ approximation work under the special cases. Cohen \textit{et al.} \cite{CohenMM17} gives a one-pass subset section that offers multiplicative $(1+\epsilon)$ approximation guarantee for the special case of $\ell_{2}$ subspace approximation. Mahabadi \textit{et al.} \cite{MahabadiRWZ20} gives a one-pass \emph{noisy} subset selection with $(1+\epsilon)$ approximation guarantee for $\ell_{p}$ subspace approximation when $p \in \{1, 2\}$. Our subset selection algorithm gives a weaker, additive approximation guarantee, but it works for any $p \in [1, \infty)$.

We consider the problem of subset selection for $\ell_{p}$ subspace approximation, i.e., given $n$ points in $d$ dimensions, we need to pick a small, representative subset of the given points such that its span gives $(1+\epsilon)$ approximation to the best $k$-dimensional subspace that minimizes the sum of $p$-th powers of distances of all the points to this subspace. Sampling-based subset selection techniques require adaptive sampling iterations with multiple passes over the data. Matrix sketching techniques give a single-pass $(1+\epsilon)$ approximation for $\ell_{p}$ subspace approximation but require additional passes for subset selection. In this work, we propose an MCMC algorithm to reduce the number of passes required by previous subset selection algorithms based on adaptive sampling. For $p=2$, our algorithm gives subset selection of nearly optimal size in only $2$ passes, whereas the number of passes required in previous work depend on $k$. Our algorithm picks a subset of size $\mathrm{poly}(k/\epsilon)$ that gives $(1+\epsilon)$ approximation to the optimal subspace. The running time of the algorithm is $nd + d~\mathrm{poly}(k/\epsilon)$. We extend our results to the case when outliers are present in the datasets, and suggest a two pass algorithm for the same. Our ideas also extend to give a reduction in the number of passes required by adaptive sampling algorithms for $\ell_{p}$ subspace approximation and subset selection, for $p \geq 2$.

We study robust subspace estimation in the streaming and distributed settings. Given a set of n data points {a_i}_{i=1}^n in R^d and an integer k, we wish to find a linear subspace S of dimension k for which sum_i M(dist(S, a_i)) is minimized, where dist(S,x) := min_{y in S} |x-y|_2, and M() is some loss function. When M is the identity function, S gives a subspace that is more robust to outliers than that provided by the truncated SVD. Though the problem is NP-hard, it is approximable within a (1+epsilon) factor in polynomial time when k and epsilon are constant. We give the first sublinear approximation algorithm for this problem in the turnstile streaming and arbitrary partition distributed models, achieving the same time guarantees as in the offline case. Our algorithm is the first based entirely on oblivious dimensionality reduction, and significantly simplifies prior methods for this problem, which held in neither the streaming nor distributed models.

We study robust subspace estimation in the streaming and distributed settings. Given a set of n data points {a_i}_{i=1}^n in R^d and an integer k, we wish to find a linear subspace S of dimension k for which sum_i M(dist(S, a_i)) is minimized, where dist(S,x) := min_{y in S} |x-y|_2, and M() is some loss function. When M is the identity function, S gives a subspace that is more robust to outliers than that provided by the truncated SVD. Though the problem is NP-hard, it is approximable within a (1+epsilon) factor in polynomial time when k and epsilon are constant. We give the first sublinear approximation algorithm for this problem in the turnstile streaming and arbitrary partition distributed models, achieving the same time guarantees as in the offline case. Our algorithm is the first based entirely on oblivious dimensionality reduction, and significantly simplifies prior methods for this problem, which held in neither the streaming nor distributed models.

Tensor network decomposition, originated from quantum physics to model entangled many-particle quantum systems, turns out to be a promising mathematical technique to efficiently represent and process big data in parsimonious manner. In this study, we show that tensor networks can systematically partition structured data, e.g. color images, for distributed storage and communication in privacy-preserving manner. Leveraging the sea of big data and metadata privacy, empirical results show that neighbouring subtensors with implicit information stored in tensor network formats cannot be identified for data reconstruction. This technique complements the existing encryption and randomization techniques which store explicit data representation at one place and highly susceptible to adversarial attacks such as side-channel attacks and de-anonymization. Furthermore, we propose a theory for adversarial examples that mislead convolutional neural networks to misclassification using subspace analysis based on singular value decomposition (SVD). The theory is extended to analyze higher-order tensors using tensor-train SVD (TT-SVD); it helps to explain the level of susceptibility of different datasets to adversarial attacks, the structural similarity of different adversarial attacks including global and localized attacks, and the efficacy of different adversarial defenses based on input transformation. An efficient and adaptive algorithm based on robust TT-SVD is then developed to detect strong and static adversarial attacks.