Sharp condition found for Burer-Monteiro method to work for MaxCut-type SDPs.
problem MaxCut-type semidefinite programs with low-rank solutions.
method Sharp condition on Laplacian matrix conditioning for global minimizers of non-convex problem.
result Any second-order critical point is a global minimizer under the given condition.
Paper shows Burer-Monteiro method can solve SDPs in polynomial time under smoothed analysis.
problem Solving large-scale semidefinite programs (SDPs) efficiently.
method Perturbing SDP to create a nonconvex program in Y where Y is an nimesp matrix. result The Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time under certain conditions.
The paper reformulates clustering as matrix factorization on the Stiefel manifold.
problem Clustering high-dimensional data like images and gene expression.
method Reformulates clustering as low-rank matrix estimation, using Burer-Monteiro factorization on the Stiefel manifold.
result Proves novel prediction bounds for clustering and proposes a componentwise Langevin sampler.
Improved guarantees for nonconvex matrix factorization with rank overparameterization.
problem Minimizing nonconvex objective over low-rank matrices.
method Overparameterized Burer--Monteiro approach, leveraging smoothness and strong convexity.
result Local optimization globally converges to global optimum under certain rank conditions.
We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-defi…
This work shows that a simple local search can recover true principal components in non-negative rank-1 RPCA.
problem Recovering true principal components in non-negative rank-1 robust principal component analysis with noisy measurements.
method Using the Burer-Monteiro approach to cast RPCA as a non-convex and non-smooth ℓ1 optimization problem. result The low-dimensional formulation of symmetric and asymmetric positive rank-1 RPCA has a unique global solution and no spurious local solutions.
A new algorithm solves semidefinite programs using Langevin diffusion.
problem Optimizing semidefinite programs with diagonal constraints.
method Langevin diffusion on a product manifold of spheres.
result Langevin algorithm achieves ε accuracy in Ω(ε^-5) iterations.
Paper develops efficient AltMin algorithm for SRPCP robust matrix recovery.
problem SRPCP model robust matrix recovery with universal penalty parameter.
method Tuning-free alternating minimization (AltMin) algorithm with closed-form subproblems.
result Efficient AltMin algorithm confirms robustness and efficiency.
We consider the non-square matrix sensing problem, under restricted isometry property (RIP) assumptions. We focus on the non-convex formulation, where any rank-r matrix X∈Rm×n is represented as UV⊤, where U∈Rm×r and V∈Rn×r. In this paper…
Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…
We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With O(μr2κ2nmax(μ,logn)) random observations of a $n_1 \times n…
Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to add…
New algorithm improves clustering accuracy without sacrificing scalability.
problem Improving clustering accuracy for large datasets.
method Nonnegative low-rank semidefinite programming with Burer-Monteiro factorization.
result Significantly smaller mis-clustering errors compared to existing methods.
Paper improves understanding of noisy matrix completion using convex relaxation and nonconvex optimization.
problem Estimating a low-rank matrix from noisy partial entries.
method Combining convex relaxation and the nonconvex Burer-Monteiro approach.
result Convex relaxation achieves near-optimal estimation errors for noisy matrix completion.
Efficiently recovers low-tubal-rank tensors from few measurements.
problem Recovering tensors with low tubal-rank from limited measurements.
method Factorization and factorized gradient descent.
result Factorized gradient descent reduces computational costs and storage requirements.
Low-rank factorization is a standard way to make structured optimization problems in machine learning more tractable by replacing matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric Burer--Monteiro factorization (sBMF) writes Z=XX⊤ with a single low-rank factor X. A r…
Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the She…
Paper develops methods for non-quadratic loss low-rank matrix recovery.
problem Recovery of low-rank matrices with non-quadratic losses.
method Projected gradient method with a regularity projection oracle.
result Projected gradient method converges globally and linearly.
APGD algorithm reconstructs point set from partial distance measurements.
problem Reconstructing point set configuration from partial Euclidean distance measurements.
method Asymmetric Projected Gradient Descent (APGD) for EDMC problem.
result Global convergence and exact recovery with O(μ2r3κ2nlogn) observations. Consider an unknown smooth function f:[0,1]d→R, and say we are given n noisy mod 1 samples of f, i.e., yi=(f(xi)+ηi)mod1, for xi∈[0,1]d, where ηi denotes the noise. Given the samples (xi,yi)i=1n, our goal is to recover smooth, robust estimates of the clean sa…
Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.
problem Finding global optima in nonconvex Burer-Monteiro factorization.
method Preconditioned gradient descent for overparameterized nonconvex function minimization.
result Gradient descent with preconditioning achieves linear convergence in the overparameterized case.
Geometric technique determines exactness of SDP robustness certificate.
problem Certifying robustness of neural networks to adversarial examples.
method Geometric projection onto hyperbola, SDP relaxation of ReLU activation.
result SDP certificate is exact for a single hidden layer under mild assumptions.
Paper tackles robust matrix completion with heavy-tailed noise.
problem Estimating a low-rank matrix from noisy incomplete data.
method Adaptive Huber loss for robustness, nonconvex algorithm with spectral initialization.
result Achieves minimax-optimal statistical estimation error under bounded second moment condition.
Simplifies solving noisy SDPs for low rank matrix recovery problems.
problem Solving SDPs with noisy data for low rank matrix recovery problems.
method Identifies conditions called simplicity to limit error in noisy SDP solutions.
result Simple SDPs can be efficiently solved and their approximate solutions trusted.
Geometric approach combines asset returns and investor views for better portfolio optimization.
problem Optimizing portfolios with investor-specific views.
method Generalized Wasserstein barycenter (GWB) to integrate statistical asset returns and investor views.
result The geometric approach offers more flexibility and rewards for correct investor views.
Paper proposes an alternative method to price American options using HJM approach.
problem Price American options efficiently and accurately.
method Utilizes HJM technique to model term structure of volatility for equity markets.
result Proposes a new value function, stopping criteria, and stopping time for American options.
We study inference and learning based on a sparse coding model with `spike-and-slab' prior. As in standard sparse coding, the model used assumes independent latent sources that linearly combine to generate data points. However, instead of using a standard sparse prior such as a Laplace distribution, we study the applic…
Quantum machine learning: Adiabatic quantum SVM outperforms classical methods.
problem Training support vector machines efficiently on large datasets.
method Adiabatic quantum computing for SVM training.
result Quantum approach outperforms classical methods in accuracy and scalability.
We develop a semi-analytic approach to the valuation of auto-callable structures with accrual features subject to barrier conditions. Our approach is based on recent studies of multi-assed binaries, present in the literature. We extend these studies to the case of time-dependent parameters. We compare numerically the s…
Two ML approaches learn local volatility surfaces from option prices, with GP being arbitrage-free.
problem Interpolating European vanilla option prices to create a local volatility surface.
method Gaussian process regression and neural net with arbitrage penalties.
result GP approach is arbitrage-free and yields best out-of-sample calibration error.
This paper critiques the Standardized Measurement Approach (SMA) for operational risk and recommends maintaining Advanced Measurement Approach (AMA).
problem Weaknesses and failures of the Standardized Measurement Approach (SMA) in operational risk.
method Critical review and analysis of SMA and AMA approaches.
result SMA is unstable, insensitive to risk, and implicitly related to systemic risk in the banking sector.
A new Euclidean approach reveals the pentagram map's beauty.
problem Exploring the pentagram map through classical geometry.
method Introducing an alternative Euclidean approach.
result Demonstrates the pentagram map's elegance through classical geometry.
Two approaches extend knowledge distillation to Gaussian Processes, showing relationships to existing methods.
problem Applying knowledge distillation to Gaussian Processes for regression and classification.
method Data-centric and distribution-centric approaches to extend distillation to GPR and GPC.
result Distribution-centric approach for GPC approximately corresponds to data duplication and scaling.
We discuss the relative merits of optimistic and randomized approaches to exploration in reinforcement learning. Optimistic approaches presented in the literature apply an optimistic boost to the value estimate at each state-action pair and select actions that are greedy with respect to the resulting optimistic value f…
Online boosting method improves weak to strong learner.
problem Online learning of weak to strong learner.
method Extends batch GentleAdaBoost to online approach with line search.
result Online boosting performs better than other methods.
Predicting potential credit default accounts in advance is challenging. Traditional statistical techniques typically cannot handle large amounts of data and the dynamic nature of fraud and humans. To tackle this problem, recent research has focused on artificial and computational intelligence based approaches. In this …
In this paper, we present a new wrapper feature selection approach based on Jensen-Shannon (JS) divergence, termed feature selection with maximum JS-divergence (FSMJ), for text categorization. Unlike most existing feature selection approaches, the proposed FSMJ approach is based on real-valued features which provide mo…
Classical approaches to isometric embedding simplified.
problem Isometric embedding of Riemannian surfaces in Euclidean 3-space.
method Coordinate-based and moving-frames approaches, focusing on integrability of PDEs.
result The integrability of the PDE is surprisingly easy and related to moving frames approach.
Two new methods for option pricing without or with a riskless asset.
problem Traditional option pricing methods require a riskless asset and may not be market-complete.
method Develops two approaches: one without a riskless asset and one with.
result Both methods produce the same option prices as classical approaches.
An integrated and extendable approach for stress-testing loan portfolios
problem Stress-testing loan portfolios
method Simulate completed portfolios, generate uncertain cash flow history, compute credit risk metrics
result Enhanced stress-testing practices within any bank
Bayesian symbolic regression automates model discovery from data.
problem Learning closed-form mathematical models from data using heuristic methods.
method Probabilistic approach to symbolic regression, connecting to information theory and statistical physics.
result Probabilistic approach provides model plausibility and performance guarantees.
Paper compares neural network approaches to Optimal Transport.
problem Learning Optimal Maps between probability distributions.
method Two categories of approaches: heuristic and math-justified. Novel approach involves dynamic flows and supervised learning.
result Novel approach involving dynamic flows and reductions of Optimal Transport to supervised learning.
New approach interprets Nyström for kernel machines with geometric insight.
problem No comparative study over Nyström-based kernel machine approaches.
method Developed a new approach with geometric interpretation, showing equivalence to existing methods.
result Proposed approach offers insights into approximation errors and accuracy.
Common Representation Learning (CRL), wherein different descriptions (or views) of the data are embedded in a common subspace, is receiving a lot of attention recently. Two popular paradigms here are Canonical Correlation Analysis (CCA) based approaches and Autoencoder (AE) based approaches. CCA based approaches learn …
Two approaches detect EV charging patterns at stations.
problem Identify charging patterns at electric vehicle charging stations.
method Two approaches: rule-based and hierarchical clustering.
result Hierarchical clustering revealed unexpected charging patterns.
Deep learning outperforms classic machine learning in DAS event detection.
problem Event detection in Distributed Acoustic Sensing (DAS).
method Comparison of classic machine learning and image-based deep learning approaches.
result Image-based deep learning offers significantly faster event detection and execution times.
Survey on methods to learn graph data representations.
problem Designing optimal Neural Network architectures for arbitrary graphs.
method Review of graph kernel methods, convolutional approaches, graph neural networks, graph embedding, and probabilistic approaches.
result Discussion of various methods for learning graph data representations.
Proposes ACP for efficient inference in noisy-or models.
problem Efficient inference in noisy-or models.
method Hybrid approach combining classical and modern variational inference.
result ACP outperforms or matches other approaches in noisy-or models.