Given a sparse rating matrix and an auxiliary matrix of users or items, how can we accurately predict missing ratings considering different data contexts of entities? Many previous studies proved that utilizing the additional information with rating data is helpful to improve the performance. However, existing methods …
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48 results for “matrix context”
New model predicts drug effects across various cell types using causal imputation.
problem Predict drug effects across different cell types given limited data.
method Introduces a novel SCM-based model class with latent factor structure and uses Synthetic Interventions estimator.
result Method outperforms other matrix completion approaches in drug repurposing dataset.
A model for POI recommendation using relation embedding.
problem Challenges in POI recommendation due to sparse user-POI matrix and varying context.
method Translation-based relation embedding using Knowledge Graph Embedding techniques, combined matrix factorization framework.
result Demonstrates effectiveness of the proposed model on real-world datasets.
New online method for statistical inference with matrix context in decision-making.
problem Statistical inference in decision-making with matrix context.
method Proposes a fully online procedure to conduct statistical inference with adaptive data collection, handling low-rank structure.
result Establishes asymptotic normality of debiased estimators and proves validity of confidence intervals.
Fine-tuning harms in-context learning, but restricting updates to the value matrix improves zero-shot performance.
problem Fine-tuning harms in-context learning, reducing zero-shot performance on unseen tasks.
method Theoretical analysis of linear attention models, identifying conditions for degraded few-shot performance.
result Restricting updates to the value matrix improves zero-shot performance while preserving in-context learning.
Particles representing tokens cluster in Transformers, influenced by initial tokens and matrix spectrum.
problem Understanding the geometry of learned representations in Transformers.
method Viewing Transformers as particle systems, applying dynamical systems and partial differential equations.
result Particles cluster towards limiting objects, confirming context-awareness and the emergence of leaders.
In this paper we propose a method for a quantitative estimation of the decision maker's knowledge in the context of the Analytic Hierarchy Process (AHP) in cases, where the judgment matrix is inconsistent. We show that the matrix of deviation from the transitivity condition corresponds to the rate matrix for transactio…
Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
Study reveals how depth of reasoning affects generalization in models.
problem Understanding scaling behavior of generalization with CoT depth.
method Theoretical model of CoT in linear regression using random matrix theory.
result Sharp phase transition between exponential and polynomial improvement, saturation, and overthinking.
New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
Recommender systems (RS), which have been an essential part in a wide range of applications, can be formulated as a matrix completion (MC) problem. To boost the performance of MC, matrix completion with side information, called inductive matrix completion (IMC), was further proposed. In real applications, the factorize…
SNN architecture shows gradient descent converges to regularized solution in matrix sensing problems.
problem Understanding implicit regularization in neural networks for matrix sensing.
method Developed Spectral Neural Networks (SNN) for matrix learning problems, rigorously demonstrating implicit regularization.
result Gradient descent converges to the solution of a regularized learning problem in matrix sensing problems.
The paper uncovers symmetries in large language models through layer-peeled optimization.
problem Understanding geometric structure in large language model weights and context embeddings.
method Constrained layer-peeled optimization program to analyze symmetries in next-token distributions.
result Symmetries in target next-token distributions are transferred to optimal model weights and context embeddings.
We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of …
From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary lear…
This paper takes a step towards theoretical analysis of the relationship between word embeddings and context embeddings in models such as word2vec. We start from basic probabilistic assumptions on the nature of word vectors, context vectors, and text generation. These assumptions are well supported either empirically o…
Incremental computation of block triangular matrix exponentials with application to option pricingmath.NA
We study the problem of computing the matrix exponential of a block triangular matrix in a peculiar way: Block column by block column, from left to right. The need for such an evaluation scheme arises naturally in the context of option pricing in polynomial diffusion models. In this setting a discretization process pro…
Eigenvector dynamics: theory and some applicationscond-mat.stat-mech
We propose a general framework to study the stability of the subspace spanned by consecutive eigenvectors of a generic symmetric matrix , when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation ( is then the Hamiltonian) and risk control …
Matrix sketching is a recently developed data compression technique. An input matrix A is efficiently approximated with a smaller matrix B, so that B preserves most of the properties of A up to some guaranteed approximation ratio. In so doing numerical operations on big data sets become faster. Sketching algorithms gen…
Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimizationcond-mat.stat-mech
We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate…
Mamba struggles with long context lengths, but spectrum scaling improves performance.
problem Mamba's performance degrades with increasing context length.
method Spectrum scaling applied to pre-trained Mamba models to improve long-context generalization.
result Spectrum scaling significantly improves performance in long-context settings.
We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the st…
Proposes a Bayesian approach for integrating multiple linked matrices.
problem Integrating multiple linked matrices for diverse applications.
method Empirical Bayes Linked Matrix Decomposition (EB-LMD).
result Efficient estimation algorithm with no tuning parameters.
We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade. This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum …
This letter proposes a dictionary learning algorithm for blind one bit compressed sensing. In the blind one bit compressed sensing framework, the original signal to be reconstructed from one bit linear random measurements is sparse in an unknown domain. In this context, the multiplication of measurement matrix $\Ab$ an…
High-dimensional time series prediction is needed in applications as diverse as demand forecasting and climatology. Often, such applications require methods that are both highly scalable, and deal with noisy data in terms of corruptions or missing values. Classical time series methods usually fall short of handling bot…
Study forecasts volatility and risk in electricity markets using matrix-HAR models.
problem Forecasting volatility and risk in electricity markets.
method Constructed a parsimonious matrix-HAR type model to estimate realized covariation and risk premia in electricity markets.
result Inclusion of longer time horizons and renewable generation information improves forecasts.
An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-…
Low-rank matrix approximations are often used to help scale standard machine learning algorithms to large-scale problems. Recently, matrix coherence has been used to characterize the ability to extract global information from a subset of matrix entries in the context of these low-rank approximations and other sampling-…
We study the adaptive estimation of copula correlation matrix for the semi-parametric elliptical copula model. In this context, the correlations are connected to Kendall's tau through a sine function transformation. Hence, a natural estimate for is the plug-in estimator with Kendall's tau statistic. We …
Topic models can provide us with an insight into the underlying latent structure of a large corpus of documents. A range of methods have been proposed in the literature, including probabilistic topic models and techniques based on matrix factorization. However, in both cases, standard implementations rely on stochastic…
Understanding and Improving Continuous Adversarial Training for LLMs via In-context Learning Theorycs.LG
This paper improves continuous adversarial training for LLMs using in-context learning theory.
problem Efficiently defending large language models (LLMs) against jailbreak attacks.
method The paper presents a theoretical analysis of continuous adversarial training (CAT) for LLMs based on in-context learning (ICL) theory, proving a robust generalization bound and proposing an improved regularization term.
result The robust generalization bound explains why CAT can defend against jailbreak prompts and shows that LLM robustness is related to embedding matrix singular values.
This letter proposes a low-computational Bayesian algorithm for noisy sparse recovery in the context of one bit compressed sensing with sensing matrix perturbation. The proposed algorithm which is called BHT-MLE comprises a sparse support detector and an amplitude estimator. The support detector utilizes Bayesian hypot…
LLMs can be tricked into recalling facts based on context clues.
problem Manipulation of LLMs' factual recall through context changes.
method Mathematical exploration of transformers' associative memory properties.
result Transformers use self-attention and value matrix for associative memory.
Optimal model selection for forecasting large collections of short time series using latent space.
problem Challenges in choosing among multiple forecasting methods for large, high-dimensional time series with limited data.
method Combining low-rank temporal matrix factorization with optimal model selection using cross-validation.
result Forecasting latent factors leads to significant performance gains compared to direct uni-variate model application.
This contribution to the proceedings of the Cracow meeting on `Applications of Random Matrix Theory' summarizes a series of studies, some old and others more recent on financial applications of Random Matrix Theory (RMT). We first review some early results in that field, with particular emphasis on the applications of …
With a simple architecture and the ability to learn meaningful word embeddings efficiently from texts containing billions of words, word2vec remains one of the most popular neural language models used today. However, as only a single embedding is learned for every word in the vocabulary, the model fails to optimally re…
The study characterizes the conditioning of the Gauss-Newton matrix in neural networks.
problem Understanding the conditioning of the Gauss-Newton matrix in neural networks.
method Theoretical analysis of the GN matrix in deep linear and ReLU networks, extending to residual connections and convolutional layers.
result Established tight bounds on the condition number of the GN matrix in neural networks.
The paper explores states of financial markets using correlation matrices and their dynamics.
problem Understanding the states of financial markets based on correlations.
method Revisits previous work and introduces recent developments in practical applications.
result Analysis of trajectories and symbolic dynamics in correlation matrix space.
Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are observed contexts and arms of the bandit. The observed context influences the reward obtained through a latent confounder var…
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly …
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
Causal Matrix Completionecon.EM
New method corrects bias in missing data for matrix completion.
problem Missing data bias in matrix completion.
method Causal model and synthetic nearest neighbors (SNN) method.
result Synthetic nearest neighbors (SNN) method provides consistent and normal estimates.
A new portfolio method uses NMF for risk budgeting, outperforming classical methods.
problem Portfolio diversification and risk management in crypto and traditional assets.
method Risk factor budgeting using convex Non-negative Matrix Factorization (NMF).
result Our method outperforms classical portfolio allocations in diversification and risk profile.
The paper improves matrix completion with auxiliary covariates using LS estimation.
problem Matrix completion with noisy data and auxiliary covariates.
method Iterative least squares estimation with statistical properties derived.
result Asymptotic normal distributions of estimators for low-rank matrix and coefficient matrix.
We find a closed-form determinant for a specific sparse covariance matrix model.
problem Finding the determinant of a specific class of sparse positive definite matrices.
method Using Fourier transform of local factors, Normal Factor Graph Duality Theorem, and Matrix Determinant Lemma.
result We derive a closed-form expression for the determinant.
In this paper, we propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex optimization and from the method of parallel tangents. However, the use of extrapola…
This paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisati…