arXiv subjects

Forest Yang

Publications and source records attributed to Forest Yang.

7 recordsLinked to original sources

arXiv2021

Data-Driven Reachability analysis and Support set Estimation with Christoffel Functions

We present algorithms for estimating the forward reachable set of a dynamical system using only a finite collection of independent and identically distributed samples. The produced estimate is the sublevel set of a function called an empirical inverse Christoffel function: empirical inverse Christoffel functions are known to provide good approximations to the support of probability distributions. In addition to reachability analysis, the same approach can be applied to general problems of estimating the support of a random variable, which has applications in data science towards detection of novelties and outliers in data sets. In applications where safety is a concern, having a guarantee of accuracy that holds on finite data sets is critical. In this paper, we prove such bounds for our algorithms under the Probably Approximately Correct (PAC) framework. In addition to applying classical Vapnik-Chervonenkis (VC) dimension bound arguments, we apply the PAC-Bayes theorem by leveraging a formal connection between kernelized empirical inverse Christoffel functions and Gaussian process regression models. The bound based on PAC-Bayes applies to a more general class of Christoffel functions than the VC dimension argument, and achieves greater sample efficiency in experiments.

eess.SY↗

arXiv2021

Data-Driven Reachability Analysis with Christoffel Functions

We present an algorithm for data-driven reachability analysis that estimates finite-horizon forward reachable sets for general nonlinear systems using level sets of a certain class of polynomials known as Christoffel functions. The level sets of Christoffel functions are known empirically to provide good approximations to the support of probability distributions: the algorithm uses this property for reachability analysis by solving a probabilistic relaxation of the reachable set computation problem. We also provide a guarantee that the output of the algorithm is an accurate reachable set approximation in a probabilistic sense, provided that a certain sample size is attained. We also investigate three numerical examples to demonstrate the algorithm's capabilities, such as providing non-convex reachable set approximations and detecting holes in the reachable set.

eess.SY↗

arXiv2020

Fairness with Overlapping Groups

In algorithmically fair prediction problems, a standard goal is to ensure the equality of fairness metrics across multiple overlapping groups simultaneously. We reconsider this standard fair classification problem using a probabilistic population analysis, which, in turn, reveals the Bayes-optimal classifier. Our approach unifies a variety of existing group-fair classification methods and enables extensions to a wide range of non-decomposable multiclass performance metrics and fairness measures. The Bayes-optimal classifier further inspires consistent procedures for algorithmically fair classification with overlapping groups. On a variety of real datasets, the proposed approach outperforms baselines in terms of its fairness-performance tradeoff.

cs.LG↗

arXiv2019

On the Consistency of Top-k Surrogate Losses

The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously understood when taking the $k$-argmax of a vector is guaranteed to return the $k$-argmax of another vector, though doing so is crucial to describe Bayes optimality; we do both tasks. Then, we define top-$k$ calibration and show it is necessary and sufficient for consistency. Based on the top-$k$ calibration analysis, we propose a class of top-$k$ calibrated Bregman divergence surrogates. Our analysis continues by showing previously proposed hinge-like top-$k$ surrogate losses are not top-$k$ calibrated and suggests no convex hinge loss is top-$k$ calibrated. On the other hand, we propose a new hinge loss which is consistent. We explore further, showing our hinge loss remains consistent under a restriction to linear functions, while cross entropy does not. Finally, we exhibit a differentiable, convex loss function which is top-$k$ calibrated for specific $k$.

cs.LG↗

arXiv2018

Minimal Effective Gibbs Ansatz (MEGA): A simple protocol for extracting an accurate thermal representation for quantum simulation

Quantum Gibbs state sampling algorithms generally suffer from either scaling exponentially with system size or requiring specific knowledge of spectral properties \textit{a priori}. Also, these algorithms require a large overhead of bath or scratch/ancilla qubits. We propose a method, termed the minimal effective Gibbs ansatz (MEGA), which uses a quantum computer to determine a minimal ensemble of pure states that accurately reproduce thermal averages of typical observables. This technique employs properties of correlation functions that can be split into a lesser and greater part; here, we primarily focus on single-particle Green's functions. When properly measured, these correlation functions provide a simple test to indicate how close a given pure state or ensemble of pure states are to providing accurate thermal expectation values. Further, we show that when properties such as the eigenstate thermalization hypothesis hold, this approach leads to accurate results with a sparse ensemble of pure states; sometimes only one suffices. We illustrate the ansatz using exact diagonalization simulations on small clusters for the Fermi-Hubbard and Hubbard-like models. Even if MEGA becomes as computationally complex as other Gibbs state samplers, it still gains an advantage due to its ease of implementation without any \textit{a priori} information about the Hamiltonian and in the efficient allocation of available qubits by eliminating bath qubits and using a minimal number of ancilla.

quant-ph↗

arXiv2018

Matching Observations to Distributions: Efficient Estimation via Sparsified Hungarian Algorithm

Suppose we are given observations, where each observation is drawn independently from one of $k$ known distributions. The goal is to match each observation to the distribution from which it was drawn. We observe that the maximum likelihood estimator (MLE) for this problem can be computed using weighted bipartite matching, even when $n$, the number of observations per distribution, exceeds one. This is achieved by instantiating $n$ duplicates of each distribution node. However, in the regime where the number of observations per distribution is much larger than the number of distributions, the Hungarian matching algorithm for computing the weighted bipartite matching requires $\mathcal O(n^3)$ time. We introduce a novel randomized matching algorithm that reduces the runtime to $\tilde{\mathcal O}(n^2)$ by sparsifying the original graph, returning the exact MLE with high probability. Next, we give statistical justification for using the MLE by bounding the excess risk of the MLE, where the loss is defined as the negative log-likelihood. We test these bounds for the case of isotropic Gaussians with equal covariances and whose means are separated by a distance $\eta$, and find (1) that $\gg \log k$ separation suffices to drive the proportion of mismatches of the MLE to 0, and (2) that the expected fraction of mismatched observations goes to zero at rate $\mathcal O({(\log k)}^2/\eta^2)$.

cs.DS↗

arXiv2018

Kernel-based Outlier Detection using the Inverse Christoffel Function

Outlier detection methods have become increasingly relevant in recent years due to increased security concerns and because of its vast application to different fields. Recently, Pauwels and Lasserre (2016) noticed that the sublevel sets of the inverse Christoffel function accurately depict the shape of a cloud of data using a sum-of-squares polynomial and can be used to perform outlier detection. In this work, we propose a kernelized variant of the inverse Christoffel function that makes it computationally tractable for data sets with a large number of features. We compare our approach to current methods on 15 different data sets and achieve the best average area under the precision recall curve (AUPRC) score, the best average rank and the lowest root mean square deviation.

stat.ML↗