arXiv research

Efficiently constructs prediction bands with minimal assumptions.

This paper proposes a variant of the method of Guédon and Verhynin for estimating the cluster matrix in the Mixture of Gaussians framework via Semi-Definite Programming. A clustering oriented embedding is deduced from this estimate. The procedure is suitable for very high dimensional data because it is based on pairwis…

Method learns SDEs from data snapshots.

Paper tackles clustering with ordinal comparisons, achieving near-optimal results.

We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…

Efficient PAC learning for contrastive linear representations is achieved.

We develop exact representations of training two-layer neural networks with rectified linear units (ReLUs) in terms of a single convex program with number of variables polynomial in the number of training samples and the number of hidden neurons. Our theory utilizes semi-infinite duality and minimum norm regularization…

New regularizer for machine learning using private data.

We consider the problem of estimating the phases of K mixed complex signals from a multichannel observation, when the mixing matrix and signal magnitudes are known. This problem can be cast as a non-convex quadratically constrained quadratic program which is known to be NP-hard in general. We propose three approaches t…

Algorithm learns halfspaces in noisy data efficiently.

The framework of Integral Quadratic Constraints (IQC) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to a semi-definite program (SDP). In the case of over-relaxed Alternating Direction Method of Multipliers (ADMM), an explicit and closed form solution to this SDP was …

SOC-ICNN expands neural network representational capacity by using conic optimization.

Paper proposes a new covariance estimator ensuring positive semi-definite matrices.

Learning representation from relative similarity comparisons, often called ordinal embedding, gains rising attention in recent years. Most of the existing methods are based on semi-definite programming (\textit{SDP}), which is generally time-consuming and degrades the scalability, especially confronting large-scale dat…

A new method solves diagonally constrained SDPs quickly and accurately.

Improves scalability of Bayesian optimization for combinatorial spaces.

The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.

Proves Gerber statistic is always non-negative.

This paper addresses a novel data science problem, prescriptive price optimization, which derives the optimal price strategy to maximize future profit/revenue on the basis of massive predictive formulas produced by machine learning. The prescriptive price optimization first builds sales forecast formulas of multiple pr…

Paper uses SDP for community detection with side information.

Generalizes leverage score sampling for neural networks, accelerating kernel methods and deep learning.

This paper describes a fast algorithm for recovering low-rank matrices from their linear measurements contaminated with Poisson noise: the Poisson noise Maximum Likelihood Singular Value thresholding (PMLSV) algorithm. We propose a convex optimization formulation with a cost function consisting of the sum of a likeliho…

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does…

New algorithm estimates task affinities without repeated training, improving model performance and efficiency.

RedEx improves neural network optimization with convex optimization guarantees.

In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated as low-rank semi-definite programming (SDP) problems. Traditional formulations us…

Due to limited metering infrastructure, distribution grids are currently challenged by observability issues. On the other hand, smart meter data, including local voltage magnitudes and power injections, are communicated to the utility operator from grid buses with renewable generation and demand-response programs. This…

Optimal neural network approximation for Wasserstein gradient direction via convex optimization.

New approach to analyze matrix denoising using gradient flow and fixed point equations.

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms which perform clustering on a number of small subgraphs and finally patches the results into a single clustering. The main advantage of these algorithms is that they bring down si…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Integrates side information for robust portfolio optimization.

The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Followup work by Nishihara et al. (2015) applies this technique to the entire family of over-relaxed Alternating…

Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this p…

Selecting hyperparameters for unsupervised learning problems is challenging in general due to the lack of ground truth for validation. Despite the prevalence of this issue in statistics and machine learning, especially in clustering problems, there are not many methods for tuning these hyperparameters with theoretical …

Paper develops Riemannian geometry for SPSD matrices with DA applications.

Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing …

A new method for deep Wishart processes improves kernel-based models.

New algorithm solves fair PCA, robust PCA, and sparse PCA problems efficiently.

Algorithm learns two-layer residual units using ReLU activations from samples.

Paper quantizes heavy-tailed data for near optimal estimation rates.

The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatori…

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistica…

Survey of kernels, RKHS, and their applications in machine learning.

In this paper we present a slight modification of the Fourier estimation method of the spot volatility (matrix) process of a continuous Itô semimartingale where the estimators are always non-negative definite. Since the estimators are factorized, computational cost will be saved a lot.

In this paper we study the support recovery problem for single index models $Y=f(\boldsymbol{X}^{\intercal} \boldsymbolβ,\varepsilon)$, where $f$ is an unknown link function, $\boldsymbol{X}\sim N_p(0,\mathbb{I}_{p})$ and $\boldsymbolβ$ is an $s$-sparse unit vector such that $\boldsymbolβ_{i}\in \{\pm\frac{1}{\sqrt{s}}…