arXiv research

Gradient descent proves global convergence for 4-layer matrix factorization.

Mixed membership factorization is a popular approach for analyzing data sets that have within-sample heterogeneity. In recent years, several algorithms have been developed for mixed membership matrix factorization, but they only guarantee estimates from a local optimum. Here, we derive a global optimization (GOP) algor…

New method detects global factors near BBP phase transition in high-dimensional data.

The study examines how global economic policy uncertainty affects crude oil futures volatility.

The study examines cross-border lending behavior from G7 countries, showing changes in driving factors after the 2008 financial crisis.

Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.

This study analyzes dynamic connectedness in global supply chain infrastructure portfolios, identifying key risk factors and extreme events.

Study uses APT and QR to identify risk factors affecting crude oil returns.

We propose a modified time lag random matrix theory in order to study time lag cross-correlations in multiple time series. We apply the method to 48 world indices, one for each of 48 different countries. We find long-range power-law cross-correlations in the absolute values of returns that quantify risk, and find that …

Techniques involving factorization are found in a wide range of applications and have enjoyed significant empirical success in many fields. However, common to a vast majority of these problems is the significant disadvantage that the associated optimization problems are typically non-convex due to a multilinear form or…

Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.

Global fixed income returns span across multiple maturities and economies, that is, they naturally reside on multi-dimensional data structures referred to as tensors. In contrast to standard "flat-view" multivariate models that are agnostic to data structure and only describe linear pairwise relationships, we introduce…

We introduce a new factor model for log volatilities that performs dimensionality reduction and considers contributions globally through the market, and locally through cluster structure and their interactions. We do not assume a-priori the number of clusters in the data, instead using the Directed Bubble Hierarchical …

GLSKF improves tensor completion by capturing both global and local variations.

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary …

Improved guarantees for nonconvex matrix factorization with rank overparameterization.

Proposes a wave-constrained matrix factorization for signal learning.

Paper checks SSC for matrix factorizations using Gurobi.

Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.

Bayesian nonparametric models for data with heterogeneous particles.

Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster tha…

Model predicts global financial market risks and asset allocation.

New risk factors improve stress testing accuracy.

Study compares CDS databases and finds discrepancies due to various factors.

We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial ti…

New heat flow for harmonic maps avoids singularities but not bubbles.

Model for dynamic relational data with regime changes.

Learning new representations of input observations in machine learning is often tackled using a factorization of the data. For many such problems, including sparse coding and matrix completion, learning these factorizations can be difficult, in terms of efficiency and to guarantee that the solution is a global minimum.…

GIV methodology extends instrumental variable estimation for high-dimensional data.

Investigates the long-only minimum variance portfolio in factor models.

New methods optimize functions on hyperbolic and spherical spaces, matching Euclidean rates up to logarithmic factors.

We study the stability and convergence of training deep ResNets with gradient descent. Specifically, we show that the parametric branch in the residual block should be scaled down by a factor $τ=O(1/\sqrt{L})$ to guarantee stable forward/backward process, where $L$ is the number of residual blocks. Moreover, we establi…

Paper shows no spurious local minima in a specific matrix factorization problem.

Mathematical models reveal key factors for engaging Gen Z at work.

This study quantifies systemic importance in global banks using a continuous framework that amplifies localized shocks.

A new method ranks and selects features without model fitting.

Reverse annealing boosts quantum matrix factorization performance.

A new portfolio method uses NMF for risk budgeting, outperforming classical methods.

Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. The proposed algorithm is guaranteed to converge to…

Gradient descent converges geometrically to optimal self-attention parameters.

We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step …

The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the mo…

We prove that for any k greater or equal to 2, given a smooth compact k-dimensional manifold and a multiplicative k-1-gerbe on a Lie group, together with an integrable connection, there is a line bundle on the corresponding Beilinson-Drinfeld Grassmannian having the factorization property. We show that taking global se…

Local mappings relate dual and primal factor graphs for efficient marginal probability estimation.

A new deep generative model captures global dependencies without supervision.

Sharp condition found for Burer-Monteiro method to work for MaxCut-type SDPs.

This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and…

Existing nonnegative matrix factorization methods focus on learning global structure of the data to construct basis and coefficient matrices, which ignores the local structure that commonly exists among data. In this paper, we propose a new type of nonnegative matrix factorization method, which learns local similarity …