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We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ and the Nuclear norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration based convex optimization methods which provide better \mmode{ O \left( \nicefrac{1}{k^2} \right) } theoretical convergence guarantee. However, to fully exploit the utility of these reformulations, we also provide a novel algorithm, to which we refer as the Fast Linear Inverse Problem Solver (FLIPS), that is tailored to solve the reformulation of the LIP. We demonstrate the performance of FLIPS on the sparse coding problem arising in image processing tasks. In this setting, we observe that FLIPS consistently outperforms the Chambolle-Pock and C-SALSA algorithms--two of the current best methods in the literature.

The Sinkhorn algorithm (arXiv:1306.0895) is the state-of-the-art to compute approximations of optimal transport distances between discrete probability distributions, making use of an entropically regularized formulation of the problem. The algorithm is guaranteed to converge, no matter its initialization. This lead to little attention being paid to initializing it, and simple starting vectors like the n-dimensional one-vector are common choices. We train a neural network to compute initializations for the algorithm, which significantly outperform standard initializations. The network predicts a potential of the optimal transport dual problem, where training is conducted in an adversarial fashion using a second, generating network. The network is universal in the sense that it is able to generalize to any pair of distributions of fixed dimension after training, and we prove that the generating network is universal in the sense that it is capable of producing any pair of distributions during training. Furthermore, we show that for certain applications the network can be used independently.

We present a new algorithm to learn a deep neural network model robust against adversarial attacks. Previous algorithms demonstrate an adversarially trained Bayesian Neural Network (BNN) provides improved robustness. We recognize the adversarial learning approach for approximating the multi-modal posterior distribution of a Bayesian model can lead to mode collapse; consequently, the model's achievements in robustness and performance are sub-optimal. Instead, we first propose preventing mode collapse to better approximate the multi-modal posterior distribution. Second, based on the intuition that a robust model should ignore perturbations and only consider the informative content of the input, we conceptualize and formulate an information gain objective to measure and force the information learned from both benign and adversarial training instances to be similar. Importantly. we prove and demonstrate that minimizing the information gain objective allows the adversarial risk to approach the conventional empirical risk. We believe our efforts provide a step toward a basis for a principled method of adversarially training BNNs. Our model demonstrate significantly improved robustness--up to 20%--compared with adversarial training and Adv-BNN under PGD attacks with 0.035 distortion on both CIFAR-10 and STL-10 datasets.

We revisit the problem of finding small $\epsilon$-separation keys introduced by Motwani and Xu (2008). In this problem, the input is $m$-dimensional tuples $x_1,x_2,\ldots,x_n $. The goal is to find a small subset of coordinates that separates at least $(1-\epsilon){n \choose 2}$ pairs of tuples. They provided a fast algorithm that runs on $\Theta(m/\epsilon)$ tuples sampled uniformly at random. We show that the sample size can be improved to $\Theta(m/\sqrt{\epsilon})$. Our algorithm also enjoys a faster running time. To obtain this result, we provide upper and lower bounds on the sample size to solve the following decision problem. Given a subset of coordinates $A$, reject if $A$ separates fewer than $(1-\epsilon){n \choose 2}$ pairs, and accept if $A$ separates all pairs. The algorithm must be correct with probability at least $1-\delta$ for all $A$. We show that for algorithms based on sampling: - $\Theta(m/\sqrt{\epsilon})$ samples are sufficient and necessary so that $\delta \leq e^{-m}$ and - $\Omega(\sqrt{\frac{\log m}{\epsilon}})$ samples are necessary so that $\delta$ is a constant. Our analysis is based on a constrained version of the balls-into-bins problem. We believe our analysis may be of independent interest. We also study a related problem that asks for the following sketching algorithm: with given parameters $\alpha,k$ and $\epsilon$, the algorithm takes a subset of coordinates $A$ of size at most $k$ and returns an estimate of the number of unseparated pairs in $A$ up to a $(1\pm\epsilon)$ factor if it is at least $\alpha {n \choose 2}$. We show that even for constant $\alpha$ and success probability, such a sketching algorithm must use $\Omega(mk \log \epsilon^{-1})$ bits of space; on the other hand, uniform sampling yields a sketch of size $\Theta(\frac{mk \log m}{\alpha \epsilon^2})$ for this purpose.

Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating R\'enyi entropies as specific examples. In particular, given a quantum oracle that prepares an $n$-dimensional quantum state $\sum_{i=1}^{n}\sqrt{p_{i}}|i\rangle$, for $\alpha>1$ and $0<\alpha<1$, our algorithm framework estimates $\alpha$-R\'enyi entropy $H_{\alpha}(p)$ to within additive error $\epsilon$ with probability at least $2/3$ using $\widetilde{\mathcal{O}}(n^{1-\frac{1}{2\alpha}}/\epsilon + \sqrt{n}/\epsilon^{1+\frac{1}{2\alpha}})$ and $\widetilde{\mathcal{O}}(n^{\frac{1}{2\alpha}}/\epsilon^{1+\frac{1}{2\alpha}})$ queries, respectively. This improves the best known dependence in $\epsilon$ as well as the joint dependence between $n$ and $1/\epsilon$. Technically, our quantum algorithms combine quantum singular value transformation, quantum annealing, and variable-time amplitude estimation. We believe that our algorithm framework is of general interest and has wide applications.

Derived from spiking neuron models via the diffusion approximation, the moment activation (MA) faithfully captures the nonlinear coupling of correlated neural variability. However, numerical evaluation of the MA faces significant challenges due to a number of ill-conditioned Dawson-like functions. By deriving asymptotic expansions of these functions, we develop an efficient numerical algorithm for evaluating the MA and its derivatives ensuring reliability, speed, and accuracy. We also provide exact analytical expressions for the MA in the weak fluctuation limit. Powered by this efficient algorithm, the MA may serve as an effective tool for investigating the dynamics of correlated neural variability in large-scale spiking neural circuits.

Permutation tests date back nearly a century to Fisher's randomized experiments, and remain an immensely popular statistical tool, used for testing hypotheses of independence between variables and other common inferential questions. Much of the existing literature has emphasized that, for the permutation p-value to be valid, one must first pick a subgroup $G$ of permutations (which could equal the full group) and then recalculate the test statistic on permuted data using either an exhaustive enumeration of $G$, or a sample from $G$ drawn uniformly at random. In this work, we demonstrate that the focus on subgroups and uniform sampling are both unnecessary for validity -- in fact, a simple random modification of the permutation p-value remains valid even when using an arbitrary distribution (not necessarily uniform) over any subset of permutations (not necessarily a subgroup). We provide a unified theoretical treatment of such generalized permutation tests, recovering all known results from the literature as special cases. Thus, this work expands the flexibility of the permutation test toolkit available to the practitioner.

We study the fine-grained complexity of graph connectivity problems in unweighted undirected graphs. Recent development shows that all variants of edge connectivity problems, including single-source-single-sink, global, Steiner, single-source, and all-pairs connectivity, are solvable in $m^{1+o(1)}$ time, collapsing the complexity of these problems into the almost-linear-time regime. While, historically, vertex connectivity has been much harder, the recent results showed that both single-source-single-sink and global vertex connectivity can be solved in $m^{1+o(1)}$ time, raising the hope of putting all variants of vertex connectivity problems into the almost-linear-time regime too. We show that this hope is impossible, assuming conjectures on finding 4-cliques. Moreover, we essentially settle the complexity landscape by giving tight bounds for combinatorial algorithms in dense graphs. There are three separate regimes: (1) all-pairs and Steiner vertex connectivity have complexity $\hat{\Theta}(n^{4})$, (2) single-source vertex connectivity has complexity $\hat{\Theta}(n^{3})$, and (3) single-source-single-sink and global vertex connectivity have complexity $\hat{\Theta}(n^{2})$. For graphs with general density, we obtain tight bounds of $\hat{\Theta}(m^{2})$, $\hat{\Theta}(m^{1.5})$, $\hat{\Theta}(m)$, respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.

The era of big data provides researchers with convenient access to copious data. However, people often have little knowledge about it. The increasing prevalence of big data is challenging the traditional methods of learning causality because they are developed for the cases with limited amount of data and solid prior causal knowledge. This survey aims to close the gap between big data and learning causality with a comprehensive and structured review of traditional and frontier methods and a discussion about some open problems of learning causality. We begin with preliminaries of learning causality. Then we categorize and revisit methods of learning causality for the typical problems and data types. After that, we discuss the connections between learning causality and machine learning. At the end, some open problems are presented to show the great potential of learning causality with data.