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1. Efficient Statistics, in High Dimensions, from Truncated Samples

   Daskalakis, Constantinos; Gouleakis, Themis; Tzamos, Christos; Zampetakis, Manolis (January 2018, 59th Annual IEEE Symposium on Foundations of Computer Science)

   We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal (μ,Σ) means a samples is only revealed if it falls in some subset S⊆ℝd; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ and covariance matrix Σ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible. 

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2. Efficient Statistics, in High Dimensions, from Truncated Samples

   Daskalakis, Constantinos; Gouleakis, Themis; Tzamos, Christos; Zampetakis, Manolis (January 2018, 59th Annual IEEE Symposium on Foundations of Computer Science)

   We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal (μ,Σ) means a samples is only revealed if it falls in some subset S⊆ℝd; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ and covariance matrix Σ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible. 

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3. Outlier Robust Multivariate Polynomial Regression

   https://doi.org/10.4230/LIPIcs.ESA.2024.12  

   Arora, Vipul; Bhattacharyya, Arnab; Boban, Mathews; Guruswami, Venkatesan; Kelman, Esty (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   We study the problem of robust multivariate polynomial regression: let p: ℝⁿ → ℝ be an unknown n-variate polynomial of degree at most d in each variable. We are given as input a set of random samples (𝐱_i,y_i) ∈ [-1,1]ⁿ × ℝ that are noisy versions of (𝐱_i,p(𝐱_i)). More precisely, each 𝐱_i is sampled independently from some distribution χ on [-1,1]ⁿ, and for each i independently, y_i is arbitrary (i.e., an outlier) with probability at most ρ < 1/2, and otherwise satisfies |y_i-p(𝐱_i)| ≤ σ. The goal is to output a polynomial p̂, of degree at most d in each variable, within an 𝓁_∞-distance of at most O(σ) from p. Kane, Karmalkar, and Price [FOCS'17] solved this problem for n = 1. We generalize their results to the n-variate setting, showing an algorithm that achieves a sample complexity of O_n(dⁿlog d), where the hidden constant depends on n, if χ is the n-dimensional Chebyshev distribution. The sample complexity is O_n(d^{2n}log d), if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most O(σ), and the run-time depends on log(1/σ). In the setting where each 𝐱_i and y_i are known up to N bits of precision, the run-time’s dependence on N is linear. We also show that our sample complexities are optimal in terms of dⁿ. Furthermore, we show that it is possible to have the run-time be independent of 1/σ, at the cost of a higher sample complexity. 

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4. Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling

   Blanca, Antonio; Chen, Zongchen; Štefankovič, Daniel; Vigoda, Eric (August 2023, COLT 2023)

   We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution μ, an ε>0, and access to sampling oracle(s) for a hidden distribution π, the goal in identity testing is to distinguish whether the two distributions μ and π are identical or are at least ε-far apart. When there is only access to full samples from the hidden distribution π, 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 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 μ, then there is an efficient identity testing algorithm for any hidden distribution π using O˜(n/ε) queries to the 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 RP=NP. We complement our results with a matching Ω(n/ε) statistical lower bound for the sample complexity of identity testing in the model. 

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5. Sample Amplification: Increasing Dataset Size even when Learning is Impossible

   Axelrod, Brian; Garg, Shivam; Sharan, Vatsal; Valiant, Gregory (January 2020, International Conference on Machine Learning (ICML))

   Given data drawn from an unknown distribution, D, to what extent is it possible to ``amplify'' this dataset and faithfully output an even larger set of samples that appear to have been drawn from D? We formalize this question as follows: an (n,m) amplification procedure takes as input n independent draws from an unknown distribution D, and outputs a set of m > n ``samples'' which must be indistinguishable from m samples drawn iid from D. We consider this sample amplification problem in two fundamental settings: the case where D is an arbitrary discrete distribution supported on k elements, and the case where D is a d-dimensional Gaussian with unknown mean, and fixed covariance matrix. Perhaps surprisingly, we show a valid amplification procedure exists for both of these settings, even in the regime where the size of the input dataset, n, is significantly less than what would be necessary to learn distribution D to non-trivial accuracy. We also show that our procedures are optimal up to constant factors. Beyond these results, we describe potential applications of sample amplification, and formalize a number of curious directions for future research. 

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