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1. Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence

   https://doi.org/10.1109/FOCS54457.2022.00019  

   Anari, Nima ; Liu, Yang P. ; Vuong, Thuy-Duong ( October 2022 , 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS))

   We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$\footnote{Throughout, $\widetilde{O}(\cdot)$ hides polylogarithmic factors in $n$.} time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. This is the first nearly-linear runtime in the output size, which is clearly optimal. For a determinantal point process on $k$-sized subsets of a ground set of $n$ elements, defined via an $n\times n$ kernel matrix, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. The time to compute just the weight of the output set is simply $\simeq k^\omega$, a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, samplingmore »from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, the domain size can be reduced to nearly linear in the output size $t=\widetilde{O}(k)$, improving the state of the art from $t= \widetilde{O}(k^2)$ for general strongly Rayleigh distributions and the more specialized $t=\widetilde{O}(k^{1.5})$ for spanning tree distributions. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of $\mu$ are known and stored in a convenient data structure. Having access to marginals is the discrete analog of having access to the mean and covariance of a continuous distribution, or equivalently knowing ``isotropy'' for the distribution, the key behind optimal samplers in the continuous setting based on the famous Kannan-Lov\'asz-Simonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures.« less
2. Local Access to Huge Random Objects Through Partial Sampling

   Biswas, Amartya Shankha ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2020 , 11th Innovations in Theoretical Computer Science Conference, ITCS 2020)

   Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries inmore »the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself.« less
3. MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retraining

   Humayun, Ahmed Imtiaz ; Balestriero, Randall ; Baraniuk, Richard ( April 2022 , The International Conference on Learning Representations (ICLR) 2022)

   Deep Generative Networks (DGNs) are extensively employed in Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and their variants to approximate the data manifold and distribution. However, training samples are often distributed non-uniformly on the manifold, due to the cost or convenience of collection. For example, the CelebA dataset contains a large fraction of smiling faces. These inconsistencies will be reproduced when sampling from the trained DGN, which is not always preferred, e.g., for fairness or data augmentation. In response, we develop MaGNET, a novel and theoretically motivated latent space sampler for any pre-trained DGN that produces samples uniformly distributed on the learned manifold. We perform a range of experiments on several datasets and DGNs, e.g., for the state-of-the-art StyleGAN2 trained on the FFHQ dataset, uniform sampling via MaGNET increases distribution precision by 4.1% and recall by 3.0% and decreases gender bias by 41.2%, without requiring labels or retraining. Since uniform sample distribution does not imply uniform semantic distribution, we also explore how semantic attributes of generated samples vary under MaGNET sampling. Colab and codes at bit.ly/magnet-sampling
4. Sampling Correctors

   Canonne, C. ; Gouleakis, T. ; Rubinfeld, R. ( July 2018 , SIAM journal on computing)

   In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make “on-the-fly” corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user. We show connections between sampling correctors, distribution learning algorithms, and distribution property testing algorithms. We show that these connections can be utilized to expand the applicability of known distribution learning and property testing algorithms as well as to achieve improved algorithms for those tasks. As a first step, we show how to design sampling correctors using proper learning algorithms. We then focus on the question of whether algorithms for sampling correctors can be more efficient in terms of sample complexity than learning algorithms for the analogous families of distributions. When correcting monotonicity, we show that this is indeed the case when also granted query access to the cumulative distribution function. We also obtain sampling correctors for monotonicity even without this stronger type of access, provided that the distribution be originally very closemore »to monotone (namely, at a distance $O(1/\log^2 n)$). In addition to that, we consider a restricted error model that aims at capturing “missing data” corruptions. In this model, we show that distributions that are close to monotone have sampling correctors that are significantly more efficient than achievable by the learning approach. We consider the question of whether an additional source of independent random bits is required by sampling correctors to implement the correction process. We show that for correcting close-to-uniform distributions and close-to-monotone distributions, no additional source of random bits is required, as the samples from the input source itself can be used to produce this randomness.« less
5. Entropic independence: optimal mixing of down-up random walks

   https://doi.org/10.1145/3519935.3520048  

   Anari, Nima ; Jain, Vishesh ; Koehler, Frederic ; Pham, Huy Tuan ; Vuong, Thuy-Duong ( June 2022 , Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $\mu$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for allmore »distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: $\mu$ is entropically independent exactly when a transformed version of the generating polynomial of $\mu$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$, and (3) nearly-linear time $\widetilde O_{\delta}(n)$ samplers for the hardcore and Ising models on $n$-node graphs that have $\delta$-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree $\Delta$ of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs.« less