An official website of the United States government
We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $$n$$-node undirected graph. We provide a randomized algorithm that, with $$O(n\epsilon^{-2})$$ queries to a degree and neighbor oracle and in $$O(n\epsilon^{-3})$$ time, estimates the spectrum up to $$\epsilon$$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $$O(n\epsilon^{-7})$$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $$\epsilon$$, a $$2^{O(\epsilon^{-1})}$$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call \emph{nuclear sparsification}. We provide an $$O(n\epsilon^{-2})$$-query and $$O(n\epsilon^{-2})$$-time algorithm for computing $$O(n\epsilon^{-2})$$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first \emph{deterministic} algorithm for spectral density estimation that scales linearly with $$n$$ (sublinear in the representation size of the graph).
more »
« less
Moments, Random Walks, and Limits for Spectrum Approximation
We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $$\epsilon$$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $$(1\pm2^{-\Omega(1/\epsilon)})$$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $$2^{O(1/\epsilon)}$$ random walks initiated at uniformly random nodes in the graph.As a strengthening of our main result, we show that improving the dependence on $$1/\epsilon$$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $$\epsilon$$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $$2^{\Omega(1/\epsilon)}$$ random walks of length $$2^{\Omega(1/\epsilon)}$$ started at random nodes.
more »
« less
- Award ID(s):
- 2045590
- PAR ID:
- 10495903
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Journal Name:
- Proceedings of Thirty Sixth Conference on Learning Theory
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well.more » « less
-
The epsilon-approximate degree, deg_epsilon(f), of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND.more » « less
-
Benito, Rosa Maria; Cherifi, Chantal; Cherifi, Hocine; Moro, Esteban; Rocha, Luis M. (Ed.)To characterize the “average” of a set of graphs, one can compute the sample Fr ́echet mean. We prove the following result: if we use the Hamming distance to compute distances between graphs, then the Fr ́echet mean of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix: an edge exists between the vertices i and j in the Fr ́echet mean graph if and only if the corresponding entry of the expected adjacency matrix is greater than 1/2. We prove that the result also holds for the sample Fr ́echet mean when the expected adjacency matrix is replaced with the sample mean adjacency matrix. This novel theoretical result has some significant practical consequences; for instance, the Fr ́echet mean of an ensemble of sparse inhomogeneous random graphs is the empty graph.more » « less
-
The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]). Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm's query complexity is close to optimal in terms of its dependence on n. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
