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In this paper, we address the problem of how many randomly labeled patterns can be correctly classified by a single-layer perceptron when the patterns are correlated with each other. In order to solve this problem, two analytical schemes…
This paper addresses biquadratic polynomial programming (BPP), an NP-hard optimization problem closely related to biquadratic tensors. We first establish several necessary and sufficient conditions for the positive semi-definiteness and…
Motivated by the fact that circular or spherical data are often much concentrated around a location $\pmb\theta$, we consider inference about $\pmb\theta$ under "high concentration" asymptotic scenarios for which the probability of any…
Knapsack problems are classic models that can formulate a wide range of applications. In this work, we deal with the Budgeted Maximum Coverage Problem (BMCP), which is a generalized 0-1 knapsack problem. Given a set of items with…
Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geq\lambda n$. We show that for any $\kappa \geq 0$, \[\mathbb{P}[s_n(A)\leq\kappa]\leq…
A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few…
Max-cut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NP-hard, a reality that has motivated researchers to develop a wealth of approximation…
Sparse linear regression is the well-studied inference problem where one is given a design matrix $\mathbf{A} \in \mathbb{R}^{M\times N}$ and a response vector $\mathbf{b} \in \mathbb{R}^M$, and the goal is to find a solution $\mathbf{x}…
We consider the Low Rank Approximation problem, where the input consists of a matrix $A \in \mathbb{R}^{n_R \times n_C}$ and an integer $k$, and the goal is to find a matrix $B$ of rank at most $k$ that minimizes $\| A - B \|_0$, which is…
Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…
In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent…
This paper considers smooth strongly convex and strongly concave (SC-SC) stochastic saddle point (SSP) problems. Suppose there is an arbitrary oracle that in expectation returns an $\epsilon$-solution in the sense of certain gaps, which can…
The hard core model in statistical physics is a probability distribution on independent sets in a graph in which the weight of any independent set I is proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show that…
This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity…
Linear Complementarity Problems (LCPs) with sufficient matrices form an important subclass of LCPs, and it remains a significant open question whether problems in this class can be solved in polynomial time. Kojima, Megiddo, Noma, and…
In this paper we consider the classical spherical perceptron problem. This problem and its variants have been studied in a great detail in a broad literature ranging from statistical physics and neural networks to computer science and pure…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix $A \in \mathbb{R}^{m \times n}$, output a value that is a lower bound on $\mathsf{disc}(A) = \min_{x…
In this work, we revisit algorithms for Tensor PCA: given an order-$r$ tensor of the form $T = G+\lambda \cdot v^{\otimes r}$ where $G$ is a random symmetric Gaussian tensor with unit variance entries and $v$ is an unknown boolean vector in…
It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient…