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Daniel M. Kane

Publications and source records attributed to Daniel M. Kane.

At least 127 records · Page 7Linked to original sources
arXiv2016

Minimal models of compact symplectic semitoric manifolds

A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus-focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where "minimal" is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of Vũ Ngoc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem.

math.SG↗
arXiv2016

Fourier-sparse interpolation without a frequency gap

We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval $[0, T]$ and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be above 1/T, the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary $k$-Fourier-sparse signals under $\ell_2$ bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in $k$ and logarithmic in the bandwidth and signal-to-noise ratio. As a special case, we get an algorithm to interpolate degree $d$ polynomials from noisy measurements, using $O(d)$ samples and increasing the noise by a constant factor in $\ell_2$.

cs.DS↗
arXiv2016

The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications

An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications: {\bf Learning Theory.} We design the first computationally efficient learning algorithm for PMDs with respect to the total variation distance. Our algorithm learns an arbitrary $(n, k)$-PMD within variation distance $ε$ using a near-optimal sample size of $\widetilde{O}_k(1/ε^2),$ and runs in time $\widetilde{O}_k(1/ε^2) \cdot \log n.$ Previously, no algorithm with a $\mathrm{poly}(1/ε)$ runtime was known, even for $k=3.$ {\bf Game Theory.} We give the first efficient polynomial-time approximation scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized anonymous games with $n$ players and $k$ strategies, our algorithm computes a well-supported $ε$-Nash equilibrium in time $n^{O(k^3)} \cdot (k/ε)^{O(k^3\log(k/ε)/\log\log(k/ε))^{k-1}}.$ The best previous algorithm for this problem had running time $n^{(f(k)/ε)^k},$ where $f(k) = Ω(k^{k^2})$, for any $k>2.$ {\bf Statistics.} We prove a multivariate central limit theorem (CLT) that relates an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance. Our new CLT strengthens the CLT of Valiant and Valiant by completely removing the dependence on $n$ in the error bound.

cs.DS↗
arXiv2016

Efficient Robust Proper Learning of Log-concave Distributions

We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis distribution that is as close as possible to the target, in total variation distance. In this work, we give the first computationally efficient algorithm for this learning problem. Our algorithm achieves the information-theoretically optimal sample size (up to a constant factor), runs in polynomial time, and is robust to model misspecification with nearly-optimal error guarantees. Specifically, we give an algorithm that, on input $n=O(1/\eps^{5/2})$ samples from an unknown distribution $f$, runs in time $\widetilde{O}(n^{8/5})$, and outputs a log-concave hypothesis $h$ that (with high probability) satisfies $\dtv(h, f) = O(\opt)+\eps$, where $\opt$ is the minimum total variation distance between $f$ and the class of log-concave distributions. Our approach to the robust proper learning problem is quite flexible and may be applicable to many other univariate distribution families.

cs.DS↗
arXiv2016

A New Approach for Testing Properties of Discrete Distributions

In this work, we give a novel general approach for distribution testing. We describe two techniques: our first technique gives sample-optimal testers, while our second technique gives matching sample lower bounds. As a consequence, we resolve the sample complexity of a wide variety of testing problems. Our upper bounds are obtained via a modular reduction-based approach. Our approach yields optimal testers for numerous problems by using a standard $\ell_2$-identity tester as a black-box. Using this recipe, we obtain simple estimators for a wide range of problems, encompassing most problems previously studied in the TCS literature, namely: (1) identity testing to a fixed distribution, (2) closeness testing between two unknown distributions (with equal/unequal sample sizes), (3) independence testing (in any number of dimensions), (4) closeness testing for collections of distributions, and (5) testing histograms. For all of these problems, our testers are sample-optimal, up to constant factors. With the exception of (1), ours are the {\em first sample-optimal testers for the corresponding problems.} Moreover, our estimators are significantly simpler to state and analyze compared to previous results. As an application of our reduction-based technique, we obtain the first {\em nearly instance-optimal} algorithm for testing equivalence between two {\em unknown} distributions. Moreover, our technique naturally generalizes to other metrics beyond the $\ell_1$-distance. Our lower bounds are obtained via a direct information-theoretic approach: Given a candidate hard instance, our proof proceeds by bounding the mutual information between appropriate random variables. While this is a classical method in information theory, prior to our work, it had not been used in distribution property testing.

cs.DS↗
arXiv2015

Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for depth-two linear threshold circuits with arbitrary weights, and depth-three majority circuits computing an explicit function. $\bullet$ We prove that for all $ε\gg \sqrt{\log(n)/n}$, the linear-time computable Andreev's function cannot be computed on a $(1/2+ε)$-fraction of $n$-bit inputs by depth-two linear threshold circuits of $o(ε^3 n^{3/2}/\log^3 n)$ gates, nor can it be computed with $o(ε^{3} n^{5/2}/\log^{7/2} n)$ wires. This establishes an average-case ``size hierarchy'' for threshold circuits, as Andreev's function is computable by uniform depth-two circuits of $o(n^3)$ linear threshold gates, and by uniform depth-three circuits of $O(n)$ majority gates. $\bullet$ We present a new function in $P$ based on small-biased sets, which we prove cannot be computed by a majority vote of depth-two linear threshold circuits with $o(n^{3/2}/\log^3 n)$ gates, nor with $o(n^{5/2}/\log^{7/2}n)$ wires. $\bullet$ We give tight average-case (gate and wire) complexity results for computing PARITY with depth-two threshold circuits; the answer turns out to be the same as for depth-two majority circuits. The key is a new random restriction lemma for linear threshold functions. Our main analytical tool is the Littlewood-Offord Lemma from additive combinatorics.

cs.CC↗
arXiv2015

Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables

We study the structure and learnability of sums of independent integer random variables (SIIRVs). For $k \in \mathbb{Z}_{+}$, a $k$-SIIRV of order $n \in \mathbb{Z}_{+}$ is the probability distribution of the sum of $n$ independent random variables each supported on $\{0, 1, \dots, k-1\}$. We denote by ${\cal S}_{n,k}$ the set of all $k$-SIIRVs of order $n$. In this paper, we tightly characterize the sample and computational complexity of learning $k$-SIIRVs. More precisely, we design a computationally efficient algorithm that uses $\widetilde{O}(k/ε^2)$ samples, and learns an arbitrary $k$-SIIRV within error $ε,$ in total variation distance. Moreover, we show that the {\em optimal} sample complexity of this learning problem is $Θ((k/ε^2)\sqrt{\log(1/ε)}).$ Our algorithm proceeds by learning the Fourier transform of the target $k$-SIIRV in its effective support. Its correctness relies on the {\em approximate sparsity} of the Fourier transform of $k$-SIIRVs -- a structural property that we establish, roughly stating that the Fourier transform of $k$-SIIRVs has small magnitude outside a small set. Along the way we prove several new structural results about $k$-SIIRVs. As one of our main structural contributions, we give an efficient algorithm to construct a sparse {\em proper} $ε$-cover for ${\cal S}_{n,k},$ in total variation distance. We also obtain a novel geometric characterization of the space of $k$-SIIRVs. Our characterization allows us to prove a tight lower bound on the size of $ε$-covers for ${\cal S}_{n,k}$, and is the key ingredient in our tight sample complexity lower bound. Our approach of exploiting the sparsity of the Fourier transform in distribution learning is general, and has recently found additional applications.

cs.DS↗
arXiv2015

Properly Learning Poisson Binomial Distributions in Almost Polynomial Time

We give an algorithm for properly learning Poisson binomial distributions. A Poisson binomial distribution (PBD) of order $n$ is the discrete probability distribution of the sum of $n$ mutually independent Bernoulli random variables. Given $\widetilde{O}(1/ε^2)$ samples from an unknown PBD $\mathbf{p}$, our algorithm runs in time $(1/ε)^{O(\log \log (1/ε))}$, and outputs a hypothesis PBD that is $ε$-close to $\mathbf{p}$ in total variation distance. The previously best known running time for properly learning PBDs was $(1/ε)^{O(\log(1/ε))}$. As one of our main contributions, we provide a novel structural characterization of PBDs. We prove that, for all $ε>0,$ there exists an explicit collection $\cal{M}$ of $(1/ε)^{O(\log \log (1/ε))}$ vectors of multiplicities, such that for any PBD $\mathbf{p}$ there exists a PBD $\mathbf{q}$ with $O(\log(1/ε))$ distinct parameters whose multiplicities are given by some element of ${\cal M}$, such that $\mathbf{q}$ is $ε$-close to $\mathbf{p}$. Our proof combines tools from Fourier analysis and algebraic geometry. Our approach to the proper learning problem is as follows: Starting with an accurate non-proper hypothesis, we fit a PBD to this hypothesis. More specifically, we essentially start with the hypothesis computed by the computationally efficient non-proper learning algorithm in our recent work~\cite{DKS15}. Our aforementioned structural characterization allows us to reduce the corresponding fitting problem to a collection of $(1/ε)^{O(\log \log(1/ε))}$ systems of low-degree polynomial inequalities. We show that each such system can be solved in time $(1/ε)^{O(\log \log(1/ε))}$, which yields the overall running time of our algorithm.

cs.DS↗
arXiv2015

Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao

We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p,\ p+h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to $k$-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang-Maynard-Tao theorem.

math.NT↗
arXiv2015

Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions

We give a general unified method that can be used for $L_1$ {\em closeness testing} of a wide range of univariate structured distribution families. More specifically, we design a sample optimal and computationally efficient algorithm for testing the equivalence of two unknown (potentially arbitrary) univariate distributions under the $\mathcal{A}_k$-distance metric: Given sample access to distributions with density functions $p, q: I \to \mathbb{R}$, we want to distinguish between the cases that $p=q$ and $\|p-q\|_{\mathcal{A}_k} \ge ε$ with probability at least $2/3$. We show that for any $k \ge 2, ε>0$, the {\em optimal} sample complexity of the $\mathcal{A}_k$-closeness testing problem is $Θ(\max\{ k^{4/5}/ε^{6/5}, k^{1/2}/ε^2 \})$. This is the first $o(k)$ sample algorithm for this problem, and yields new, simple $L_1$ closeness testers, in most cases with optimal sample complexity, for broad classes of structured distributions.

cs.DS↗
arXiv2015

Central Limit Theorems for some Set Partition Statistics

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

math.CO↗
arXiv2015

The Average Sensitivity of an Intersection of Half Spaces

We prove new bounds on the average sensitivity of the indicator function of an intersection of $k$ halfspaces. In particular, we prove the optimal bound of $O(\sqrt{n\log(k)})$. This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces.

cs.CC↗
arXiv2014

Testing Identity of Structured Distributions

We study the question of identity testing for structured distributions. More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$, we wish to distinguish whether $q=p$ versus $q$ is at least $ε$-far from $p$, in $L_1$ distance. In this work, we present a unified approach that yields new, simple testers, with sample complexity that is information-theoretically optimal, for broad classes of structured distributions, including $t$-flat distributions, $t$-modal distributions, log-concave distributions, monotone hazard rate (MHR) distributions, and mixtures thereof.

cs.DS↗
arXiv2014

On the Number of ABC Solutions with Restricted Radical Sizes

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture is equivalent to the statement that for $a+b+c<1$, the number of solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1 \pm ε}.$ We prove this conjecture as long as $a+b+c \geq 2.$

math.NT↗
arXiv2014

Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.

math.NT↗
arXiv2014

Sparser Johnson-Lindenstrauss Transforms

We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion $1+\varepsilon$ with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar, and Sarlós (STOC 2010). Such distributions can be used to speed up applications where $\ell_2$ dimensionality reduction is used.

cs.DS↗