Optimal estimation of Gaussian (poly)trees

We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.

Inconsistency of cross-validation for structure learning in Gaussian graphical models

Despite numerous years of research into the merits and trade-offs of various model selection criteria, obtaining robust results that elucidate the behavior of cross-validation remains a challenging endeavor. In this paper, we highlight the inherent limitations of cross-validation when employed to discern the structure of a Gaussian graphical model. We provide finite-sample bounds on the probability that the Lasso estimator for the neighborhood of a node within a Gaussian graphical model, optimized using a prediction oracle, misidentifies the neighborhood. Our results pertain to both undirected and directed acyclic graphs, encompassing general, sparse covariance structures. To support our theoretical findings, we conduct an empirical investigation of this inconsistency by contrasting our outcomes with other commonly used information criteria through an extensive simulation study. Given that many algorithms designed to learn the structure of graphical models require hyperparameter selection, the precise calibration of this hyperparameter is paramount for accurately estimating the inherent structure. Consequently, our observations shed light on this widely recognized practical challenge.

On Mergable Coresets for Polytope Distance

We show that a constant-size constant-error coreset for polytope distance is simple to maintain under merges of coresets. However, increasing the size cannot improve the error bound significantly beyond that constant.

Learning Mixtures of Gaussians with Censored Data

We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $$\sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2),$$ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $\mu_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $\mu_i$ within $\varepsilon$ error.

A super-polynomial lower bound for learning nonparametric mixtures

We study the problem of learning nonparametric distributions in a finite mixture, and establish a super-polynomial lower bound on the sample complexity of learning the component distributions in such models. Namely, we are given i.i.d. samples from $f$ where $$ f=\sum_{i=1}^k w_i f_i, \quad\sum_{i=1}^k w_i=1, \quad w_i>0 $$ and we are interested in learning each component $f_i$. Without any assumptions on $f_i$, this problem is ill-posed. In order to identify the components $f_i$, we assume that each $f_i$ can be written as a convolution of a Gaussian and a compactly supported density $\nu_i$ with $\text{supp}(\nu_i)\cap \text{supp}(\nu_j)=\emptyset$. Our main result shows that $\Omega((\frac{1}{\varepsilon})^{C\log\log \frac{1}{\varepsilon}})$ samples are required for estimating each $f_i$. The proof relies on a fast rate for approximation with Gaussians, which may be of independent interest. This result has important implications for the hardness of learning more general nonparametric latent variable models that arise in machine learning applications.

Optimal estimation of Gaussian DAG models

We study the optimal sample complexity of learning a Gaussian directed acyclic graph (DAG) from observational data. Our main result establishes the minimax optimal sample complexity for learning the structure of a linear Gaussian DAG model with equal variances to be $n\asymp q\log(d/q)$, where $q$ is the maximum number of parents and $d$ is the number of nodes. We further make comparisons with the classical problem of learning (undirected) Gaussian graphical models, showing that under the equal variance assumption, these two problems share the same optimal sample complexity. In other words, at least for Gaussian models with equal error variances, learning a directed graphical model is not more difficult than learning an undirected graphical model. Our results also extend to more general identification assumptions as well as subgaussian errors.

New Nearly-Optimal Coreset for Kernel Density Estimation

Given a point set $P\subset \mathbb{R}^d$, kernel density estimation for Gaussian kernel is defined as $\overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\in P}e^{-\left\lVert x-p \right\rVert^2}$ for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimation of $P$ can be approximated by the kernel density estimation of $Q$. This subset $Q$ is called \emph{coreset}. The primary technique in this work is to construct $\pm 1$ coloring on the point set $P$ by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyk's Theorem. When $d>1$ is constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\sqrt{\log\log\frac{1}{\varepsilon}}\right)$ as opposed to the best-known result of $O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$. It is the first to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.

Approximate Guarantees for Dictionary Learning

In the dictionary learning (or sparse coding) problem, we are given a collection of signals (vectors in $\mathbb{R}^d$), and the goal is to find a "basis" in which the signals have a sparse (approximate) representation. The problem has received a lot of attention in signal processing, learning, and theoretical computer science. The problem is formalized as factorizing a matrix $X (d \times n)$ (whose columns are the signals) as $X = AY$, where $A$ has a prescribed number $m$ of columns (typically $m \ll n$), and $Y$ has columns that are $k$-sparse (typically $k \ll d$). Most of the known theoretical results involve assuming that the columns of the unknown $A$ have certain incoherence properties, and that the coefficient matrix $Y$ has random (or partly random) structure. The goal of our work is to understand what can be said in the absence of such assumptions. Can we still find $A$ and $Y$ such that $X \approx AY$? We show that this is possible, if we allow violating the bounds on $m$ and $k$ by appropriate factors that depend on $k$ and the desired approximation. Our results rely on an algorithm for what we call the threshold correlation problem, which turns out to be related to hypercontractive norms of matrices. We also show that our algorithmic ideas apply to a setting in which some of the columns of $X$ are outliers, thus giving similar guarantees even in this challenging setting.

Relative Error RKHS Embeddings for Gaussian Kernels

We show how to obliviously embed into the reproducing kernel Hilbert space associated with Gaussian kernels, so that distance in this space (the kernel distance) only has $(1+\varepsilon)$-relative error. This only holds in comparing any point sets at a kernel distance at least $\alpha$; this parameter only shows up as a poly-logarithmic factor of the dimension of an intermediate embedding, but not in the final embedding. The main insight is to effectively modify the well-traveled random Fourier features to be slightly biased and have higher variance, but so they can be defined as a convolution over the function space. This result provides the first guaranteed algorithmic results for LSH of kernel distance on point sets and low-dimensional shapes and distributions, and for relative error bounds on the kernel two-sample test.

Near-Optimal Coresets of Kernel Density Estimates

We construct near-optimal coresets for kernel density estimate for points in $\mathbb{R^d}$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d\log (1/\epsilon)}/\epsilon)$, and we show a near-matching lower bound of size $\Omega(\sqrt{d}/\epsilon)$. The upper bound is a polynomial in $1/\epsilon$ improvement when $d \in [3,1/\epsilon^2)$ (for all kernels except the Gaussian kernel which had a previous upper bound of $O((1/\epsilon) \log^d (1/\epsilon))$) and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.