Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling
- URL: http://arxiv.org/abs/2207.09102v3
- Date: Fri, 30 Aug 2024 16:03:53 GMT
- Title: Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling
- Authors: Antonio Blanca, Zongchen Chen, Daniel Štefankovič, Eric Vigoda,
- Abstract summary: We study the identity testing problem for high-dimensional distributions.
We consider a significantly weaker conditional sampling oracle, which we call the $mathsfCoordinate Oracle$.
We prove that if an analytic property known as approximate tensorization of entropy holds for an $n$-dimensional visible distribution $mu$, then there is an efficient identity testing algorithm for any hidden distribution $pi$ using $tildeO(n/varepsilon)$ queries.
- Score: 5.3098033683382155
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution $\mu$, an $\varepsilon>0$, and access to sampling oracle(s) for a hidden distribution $\pi$, the goal in identity testing is to distinguish whether the two distributions $\mu$ and $\pi$ are identical or are at least $\varepsilon$-far apart. When there is only access to full samples from the hidden distribution $\pi$, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various "conditional" sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the $\mathsf{Coordinate\ Oracle}$, and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an $n$-dimensional visible distribution $\mu$, then there is an efficient identity testing algorithm for any hidden distribution $\pi$ using $\tilde{O}(n/\varepsilon)$ queries to the $\mathsf{Coordinate\ Oracle}$. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of $n$-dimensional distributions, specifically sparse antiferromagnetic Ising models over $\{+1,-1\}^n$, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless $\mathsf{RP}=\mathsf{NP}$. We complement our results with a matching $\Omega(n/\varepsilon)$ statistical lower bound for the sample complexity of identity testing in the $\mathsf{Coordinate\ Oracle}$ model.
Related papers
- Dimension-free Private Mean Estimation for Anisotropic Distributions [55.86374912608193]
Previous private estimators on distributions over $mathRd suffer from a curse of dimensionality.
We present an algorithm whose sample complexity has improved dependence on dimension.
arXiv Detail & Related papers (2024-11-01T17:59:53Z) - Convergence Analysis of Probability Flow ODE for Score-based Generative Models [5.939858158928473]
We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives.
We prove the total variation between the target and the generated data distributions can be bounded above by $mathcalO(d3/4delta1/2)$ in the continuous time level.
arXiv Detail & Related papers (2024-04-15T12:29:28Z) - Transfer Operators from Batches of Unpaired Points via Entropic
Transport Kernels [3.099885205621181]
We derive a maximum-likelihood inference functional, propose a computationally tractable approximation and analyze their properties.
We prove a $Gamma$-convergence result showing that we can recover the true density from empirical approximations as the number $N$ of blocks goes to infinity.
arXiv Detail & Related papers (2024-02-13T12:52:41Z) - Testing Closeness of Multivariate Distributions via Ramsey Theory [40.926523210945064]
We investigate the statistical task of closeness (or equivalence) testing for multidimensional distributions.
Specifically, given sample access to two unknown distributions $mathbf p, mathbf q$ on $mathbb Rd$, we want to distinguish between the case that $mathbf p=mathbf q$ versus $|mathbf p-mathbf q|_A_k > epsilon$.
Our main result is the first closeness tester for this problem with em sub-learning sample complexity in any fixed dimension.
arXiv Detail & Related papers (2023-11-22T04:34:09Z) - Identification of Mixtures of Discrete Product Distributions in
Near-Optimal Sample and Time Complexity [6.812247730094931]
We show, for any $ngeq 2k-1$, how to achieve sample complexity and run-time complexity $(1/zeta)O(k)$.
We also extend the known lower bound of $eOmega(k)$ to match our upper bound across a broad range of $zeta$.
arXiv Detail & Related papers (2023-09-25T09:50:15Z) - Random matrices in service of ML footprint: ternary random features with
no performance loss [55.30329197651178]
We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$.
We propose a novel random technique, called Ternary Random Feature (TRF)
The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
arXiv Detail & Related papers (2021-10-05T09:33:49Z) - The Sample Complexity of Robust Covariance Testing [56.98280399449707]
We are given i.i.d. samples from a distribution of the form $Z = (1-epsilon) X + epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $mathcalN(0, Sigma)$.
In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples.
We prove a sample complexity lower bound of $Omega(d2)$ for $epsilon$ an arbitrarily small constant and $gamma
arXiv Detail & Related papers (2020-12-31T18:24:41Z) - A Randomized Algorithm to Reduce the Support of Discrete Measures [79.55586575988292]
Given a discrete probability measure supported on $N$ atoms and a set of $n$ real-valued functions, there exists a probability measure that is supported on a subset of $n+1$ of the original $N$ atoms.
We give a simple geometric characterization of barycenters via negative cones and derive a randomized algorithm that computes this new measure by "greedy geometric sampling"
We then study its properties, and benchmark it on synthetic and real-world data to show that it can be very beneficial in the $Ngg n$ regime.
arXiv Detail & Related papers (2020-06-02T16:38:36Z) - Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
We consider the problem of learning the best-fitting single neuron as measured by the expected square loss.
For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
arXiv Detail & Related papers (2020-05-29T07:20:35Z) - Locally Private Hypothesis Selection [96.06118559817057]
We output a distribution from $mathcalQ$ whose total variation distance to $p$ is comparable to the best such distribution.
We show that the constraint of local differential privacy incurs an exponential increase in cost.
Our algorithms result in exponential improvements on the round complexity of previous methods.
arXiv Detail & Related papers (2020-02-21T18:30:48Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.