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We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection…
We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result…
The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class…
Given a Boolean function $f$ provided as a black-box with $n$ variables, this paper will propose a quantum algorithm for testing if a certain variable is junta or $\epsilon$-far from being junta. The proposed algorithm constructs another…
A function f : {0, 1}^n -> {0, 1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in…
For the multi-sample equal covariance function (ECF) testing problem, Zhang (2013) proposed an $L^{2}$-norm based test. However, its asymptotic power and finite sample performance have not been studied. In this paper, its asymptotic power…
We consider goodness-of-fit tests with i.i.d. samples generated from a categorical distribution $(p_1,...,p_k)$. For a given $(q_1,...,q_k)$, we test the null hypothesis whether $p_j=q_{\pi(j)}$ for some label permutation $\pi$. The…
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form $f(I_1(u),\ldots,I_k(u))\geq 0$ where $f$ is a polynomial, $u$ is any function satisfying prescribed…
We investigate properties of a bootstrap-based methodology for testing hypotheses about equality of certain characteristics of the distributions between different populations in the context of functional data. The suggested testing…
Most existing methods for testing equality of means of functional data from multiple populations rely on assumptions of equal covariance and/or Gaussianity. In this work we provide a new testing method based on a statistic that is…
We consider the problem of testing whether pairs of univariate random variables are associated. Few tests of independence exist that are consistent against all dependent alternatives and are distribution free. We propose novel tests that…
This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to $1/2$. Our contributions include the following: i) An analysis of the…
Testing to see whether a given data set comes from some specified distribution is among the oldest types of problems in Statistics. Many such tests have been developed and their performance studied. The general result has been that while a…
We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard…
Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general…
A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using…
We consider the convolution model where i.i.d. random variables $X_i$ having unknown density $f$ are observed with additive i.i.d. noise, independent of the $X$'s. We assume that the density $f$ belongs to either a Sobolev class or a class…
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized…
We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals…
Given a black-box representing an unknown Boolean function $f$ of $n$ variables, in this paper we propose a fast quantum algorithm to test whether or not a certain variable in the function $f$ is a junta variable. The proposed algorithm…