arXiv research

Extends phase retrieval methods to handle sensing vector errors.

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effe…

In the total least squares problem, one is given an $m \times n$ matrix $A$, and an $m \times d$ matrix $B$, and one seeks to "correct" both $A$ and $B$, obtaining matrices $\hat{A}$ and $\hat{B}$, so that there exists an $X$ satisfying the equation $\hat{A}X = \hat{B}$. Typically the problem is overconstrained, meanin…

Systematic trading strategies are rule-based procedures which choose portfolios and allocate assets. In order to attain certain desired return profiles, quantitative strategists must determine a large array of trading parameters. Backtesting, the attempt to identify the appropriate parameters using historical data avai…

Estimates manifold dimension using local graph structure.

We study the total least squares (TLS) problem that generalizes least squares regression by allowing measurement errors in both dependent and independent variables. TLS is widely used in applied fields including computer vision, system identification and econometrics. The special case when all dependent and independent…

Estimates parameters of interconnected linear systems using total variation penalization.

We consider the exploration-exploitation dilemma in finite-horizon reinforcement learning (RL). When the state space is large or continuous, traditional tabular approaches are unfeasible and some form of function approximation is mandatory. In this paper, we introduce an optimistically-initialized variant of the popula…

Study on learning sparse fixed-structure Gaussian Bayesian networks with near-optimal sample complexity.

Characterizes no Butterfly arbitrage in SVI model parameters.

Develops statistical guarantees for neural networks with regularization.

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the t…

CD converges linearly for MCP/SCAD penalized least squares.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the responses for a small subset of the points that are then used to construct linear pre…

Illustrates interleaved learning with Kalman Filter for linear least squares.

We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance meas…

Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual …

We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that …

The paper improves Kaczmarz algorithm with momentum for linear least squares.

New algorithm improves online binary classification with constant time complexity.

Reduced-rank method improves least-squares regression under output regularity.

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 …

Proposes a partitioned least squares model for feature grouping.

ESNs trained with Tikhonov least squares approximate ergodic dynamical systems in L2(μ) norm.

The kernel least mean squares (KLMS) algorithm is a computationally efficient nonlinear adaptive filtering method that "kernelizes" the celebrated (linear) least mean squares algorithm. We demonstrate that the least mean squares algorithm is closely related to the Kalman filtering, and thus, the KLMS can be interpreted…

A new algorithm solves nonnegative least squares faster with nonnegative data.

The paper identifies saddlepoints in unsupervised auto-encoding neural nets.

The paper proposes a least squares method for binary compressive sampling with low intrinsic dimension signals.

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine para…

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularize…

We propose a new forward-backward stochastic differential equation solver for high-dimensional derivatives pricing problems by combining deep learning solver with least square regression technique widely used in the least square Monte Carlo method for the valuation of American options. Our numerical experiments demonst…

We introduce a novel semi-supervised version of the least squares classifier. This implicitly constrained least squares (ICLS) classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-supervised method…

New method speeds up solving L0-regularized least-squares problems.

Least squares estimator fails to achieve optimal risk in bounded distributions, but non-linear predictors can.

Sparse linear regression, which entails finding a sparse solution to an underdetermined system of linear equations, can formally be expressed as an $l_0$-constrained least-squares problem. The Orthogonal Least-Squares (OLS) algorithm sequentially selects the features (i.e., columns of the coefficient matrix) to greedil…

In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family of such minimal surfaces with least total absolute curvature is given. Moreover, we obtain a uniqueness theorem for this family from its symmetries.

We introduce the implicitly constrained least squares (ICLS) classifier, a novel semi-supervised version of the least squares classifier. This classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-s…

We prove the statistical consistency of kernel Partial Least Squares Regression applied to a bounded regression learning problem on a reproducing kernel Hilbert space. Partial Least Squares stands out of well-known classical approaches as e.g. Ridge Regression or Principal Components Regression, as it is not defined as…

Randomized matrix compression techniques, such as the Johnson-Lindenstrauss transform, have emerged as an effective and practical way for solving large-scale problems efficiently. With a focus on computational efficiency, however, forsaking solutions quality and accuracy becomes the trade-off. In this paper, we investi…

This book introduces linear models and their theories rigorously.

The ratio of two probability densities can be used for solving various machine learning tasks such as covariate shift adaptation (importance sampling), outlier detection (likelihood-ratio test), and feature selection (mutual information). Recently, several methods of directly estimating the density ratio have been deve…

The least-squares support vector machine is a frequently used kernel method for non-linear regression and classification tasks. Here we discuss several approximation algorithms for the least-squares support vector machine classifier. The proposed methods are based on randomized block kernel matrices, and we show that t…

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.

Efficiently estimates private least squares with linear error growth.

New algorithm converts data into sub-gaussian designs efficiently.

Here, we provide a supplementary material for Takayuki Osogami, "Uncorrected least-squares temporal difference with lambda-return," which appears in {\it Proceedings of the 34th AAAI Conference on Artificial Intelligence} (AAAI-20).

We prove strong consistency and asymptotic normality of least squares estimators for the subcritical Heston model based on continuous time observations. We also present some numerical illustrations of our results.