arXiv research

We improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. 1) In compressed sensing, we show that if the m \times n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tr…

Study on market entry timing in stock liquidation with trading constraints.

New algorithm optimizes positions of CountSketch non-zero entries for better data compression.

Study transitions between tableau and spider bases for Specht modules.

A matrix completion problem, which aims to recover a complete matrix from its partial observations, is one of the important problems in the machine learning field and has been studied actively. However, there is a discrepancy between the mainstream problem setting, which assumes continuous-valued observations, and some…

New method for symmetric matrix completion using ReLU sampling.

We consider the problem of reconstructing a low rank matrix from a subset of its entries and analyze two variants of the so-called Alternating Minimization algorithm, which has been proposed in the past. We establish that when the underlying matrix has rank $r=1$, has positive bounded entries, and the graph $\mathcal{G…

Given a matrix M of low-rank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the `Netflix problem') to structure-from-motion and positioning. We study a low complexity algorithm…

The paper finds a surprising positive correlation between upstreamness and downstreamness in global value chains.

New solutions found for elliptic systems with mixed couplings.

A determinantal point process (DPP) is a probabilistic model of set diversity compactly parameterized by a positive semi-definite kernel matrix. To fit a DPP to a given task, we would like to learn the entries of its kernel matrix by maximizing the log-likelihood of the available data. However, log-likelihood is non-co…

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is derived and its conne…

Optimal timing strategy for mean-reverting price spreads.

This work generalizes transformer attention to capture higher-order correlations efficiently.

Study of a generalized geometric Brownian motion with varying entry and exit rates.

RPCholesky approximates kernel matrices with few evaluations.

New research shows larger language models improve data processing for diverse entries.

Motivated by the industry practice of pairs trading, we study the optimal timing strategies for trading a mean-reverting price spread. An optimal double stopping problem is formulated to analyze the timing to start and subsequently liquidate the position subject to transaction costs. Modeling the price spread by an Orn…

We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless…

For any matrix A in R^(m x n) of rank ρ, we present a probability distribution over the entries of A (the element-wise leverage scores of equation (2)) that reveals the most influential entries in the matrix. From a theoretical perspective, we prove that sampling at most s = O ((m + n) ρ^2 ln (m + n)) entries of the ma…

Improved matrix completion for non-uniformly sampled data.

Let $\mathfrak{S}(\underline{s},w)$ be the graph whose vertices are all subexpressions with target $w$ of a fixed expression $\underline{s}$ in generators of a Coxeter group and edges are the pairs of subexpressions with Hamming distance 2. We prove that $\mathfrak{S}(\underline{s},w)$ is connected and its cycle space …

Develops a new algorithm to calibrate signed datasets to specified marginals.

Study on optimal bubble riding with price-dependent entry times in a mean field game model.

The Burau representation of braid group B4 is shown to be faithful almost everywhere.

The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More s…

Nowadays, organizations collect vast quantities of accounting relevant transactions, referred to as 'journal entries', in 'Enterprise Resource Planning' (ERP) systems. The aggregation of those entries ultimately defines an organization's financial statement. To detect potential misstatements and fraud, international au…

The paper tackles optimal level set estimation in crowdsourcing and tournaments.

The Sinkhorn-Knopp algorithm converges quickly but the number of iterations is poorly understood.

Estimates missing distributions using nearest neighbors with kernel methods.

Paper improves tensor completion by reducing sample entries needed.

Reducing barriers to entry in large-scale ML markets, study shows multi-objective learning can lower data requirements.

A new tensor completion method handles missing data with missing not at random entries.

American Depositary Receipts (ADRs) are exchange-traded certificates that rep- resent shares of non-U.S. company securities. They are major financial instruments for investing in foreign companies. Focusing on Asian ADRs in the context of asyn- chronous markets, we present methodologies and results of empirical analysi…

New pivoting strategy improves trace norm contraction in low-rank approximation.

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we p…

Groups of matrices with integer-like entries are studied.

The completion of low rank matrices from few entries is a task with many practical applications. We consider here two aspects of this problem: detectability, i.e. the ability to estimate the rank $r$ reliably from the fewest possible random entries, and performance in achieving small reconstruction error. We propose a …

Estimates low-rank distributional matrices from incomplete samples.

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial …

Scalable and robust TR decomposition for large-scale data with missing entries and outliers.

In this paper, we consider the streaming memory-limited matrix completion problem when the observed entries are noisy versions of a small random fraction of the original entries. We are interested in scenarios where the matrix size is very large so the matrix is very hard to store and manipulate. Here, columns of the o…

We give an algorithm for completing an order-$m$ symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic result…

The covariance matrix of a $p$-dimensional random variable is a fundamental quantity in data analysis. Given $n$ i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of $O(np^{2})$ operations. When $n,p$ are large, this computation may be prohibitively slow. Moreover, …

The $\ell$-1 norm based optimization is widely used in signal processing, especially in recent compressed sensing theory. This paper studies the solution path of the $\ell$-1 norm penalized least-square problem, whose constrained form is known as Least Absolute Shrinkage and Selection Operator (LASSO). A solution path …

A new method for streaming PCA provides confidence intervals for eigenvector entries.