Yizhi Huang

Xi Chen, Anindya De, Yizhi Huang et al.

The problem of super-resolution, roughly speaking, is to reconstruct an unknown signal to high accuracy, given (potentially noisy) information about its low-degree Fourier coefficients. Prior results on super-resolution have imposed strong modeling assumptions on the signal, typically requiring that it is a linear combination of spatially separated point sources. In this work we analyze a very general version of the super-resolution problem by considering completely general non-negative signals (equivalently, distributions) over the $d$-dimensional torus $[0,1)^d$; we do not assume any spatial separation between point sources, or even that the distribution is a finite linear combination of point sources. The question naturally arises: what can be said about super-resolution in such a general setting? - As a warm-up, we first give a set of results for reconstructing distributions under the Wasserstein distance. We establish essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible: we show that for $d$-dimensional distributions, estimates of $\approx \exp(d)$ many Fourier coefficients are both necessary and sufficient for accurate Wasserstein reconstruction. - As our main result, we define a new notion of "heavy hitter" reconstruction for distributions, which essentially amounts to achieving high-accuracy reconstruction of all "sufficiently dense" regions of the distribution. We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible under this notion. Our results show that (in sharp contrast with Wasserstein reconstruction) accurate estimates of only $\approx \exp(\sqrt{d})$ many Fourier coefficients are both necessary and sufficient for heavy hitter reconstruction.

Guy Blanc, Yizhi Huang, Tal Malkin et al.

We consider the relative abilities and limitations of computationally efficient algorithms for learning in the presence of noise, under two well-studied and challenging adversarial noise models for learning Boolean functions: malicious noise, in which an adversary can arbitrarily corrupt a random subset of examples given to the learner; and nasty noise, in which an adversary can arbitrarily corrupt an adversarially chosen subset of examples given to the learner. We consider both the distribution-independent and fixed-distribution settings. Our main results highlight a dramatic difference between these two settings: For distribution-independent learning, we prove a strong equivalence between the two noise models: If a class ${\cal C}$ of functions is efficiently learnable in the presence of $η$-rate malicious noise, then it is also efficiently learnable in the presence of $η$-rate nasty noise. In sharp contrast, for the fixed-distribution setting we show an arbitrarily large separation: Under a standard cryptographic assumption, for any arbitrarily large value $r$ there exists a concept class for which there is a ratio of $r$ between the rate $η_{malicious}$ of malicious noise that polynomial-time learning algorithms can tolerate, versus the rate $η_{nasty}$ of nasty noise that such learning algorithms can tolerate. To offset the negative result for the fixed-distribution setting, we define a broad and natural class of algorithms, namely those that ignore contradictory examples (ICE). We show that for these algorithms, malicious noise and nasty noise are equivalent up to a factor of two in the noise rate: Any efficient ICE learner that succeeds with $η$-rate malicious noise can be converted to an efficient learner that succeeds with $η/2$-rate nasty noise. We further show that the above factor of two is necessary, again under a standard cryptographic assumption.

Xi Chen, Anindya De, Yizhi Huang et al.

Several recent works [DHLNSY25, CPPS25a, CPPS25b] have studied a model of property testing of Boolean functions under a \emph{relative-error} criterion. In this model, the distance from a target function $f: \{0,1\}^n \to \{0,1\}$ that is being tested to a function $g$ is defined relative to the number of inputs $x$ for which $f(x)=1$; moreover, testing algorithms in this model have access both to a black-box oracle for $f$ and to independent uniform satisfying assignments of $f$. The motivation for this model is that it provides a natural framework for testing \emph{sparse} Boolean functions that have few satisfying assignments, analogous to well-studied models for property testing of sparse graphs. The main result of this paper is a lower bound for testing \emph{halfspaces} (i.e., linear threshold functions) in the relative error model: we show that $\tildeΩ(\log n)$ oracle calls are required for any relative-error halfspace testing algorithm over the Boolean hypercube $\{0,1\}^n$. This stands in sharp contrast both with the constant-query testability (independent of $n$) of halfspaces in the standard model [MORS10], and with the positive results for relative-error testing of many other classes given in [DHLNSY25, CPPS25a, CPPS25b]. Our lower bound for halfspaces gives the first example of a well-studied class of functions for which relative-error testing is provably more difficult than standard-model testing.

Xi Chen, Anindya De, Yizhi Huang et al.

The relative-error property testing model was introduced in [CDHLNSY24] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function $f$ that is being tested to a function $g$ is defined as $\mathrm{Vol}(f \mathop{\triangle} g)/\mathrm{Vol}(f)$, where the numerator is the fraction of inputs on which $f$ and $g$ disagree and the denominator is the fraction of inputs that satisfy $f$. Recent work [CDHNSY26] has shown that over the Boolean domain $\{0,1\}^n$, any relative-error testing algorithm for the fundamental class of halfspaces (i.e. linear threshold functions) must make $Ω(\log n)$ oracle calls. In this paper we complement the [CDHNSY26] lower bound by showing that halfspaces can be relative-error tested over $\mathbb{R}^n$ under the standard $N(0,I_n)$ Gaussian distribution using a sublinear number of oracle calls -- in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area.