Daniel Štefankovič
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View article: Optimal mixing via tensorization for random independent sets on arbitrary trees
Optimal mixing via tensorization for random independent sets on arbitrary trees Open
We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more…
View article: Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling
Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling Open
We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution \(\mu\) , an \(\varepsilon \gt 0\) , and access to sampling oracle(s) for a hidden distribution \(\pi\) , the goal in identit…
View article: Fast Sampling via Spectral Independence Beyond Bounded-degree Graphs
Fast Sampling via Spectral Independence Beyond Bounded-degree Graphs Open
Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs fo…
View article: Spectral Independence and Local-to-Global Techniques for Optimal Mixing of Markov Chains
Spectral Independence and Local-to-Global Techniques for Optimal Mixing of Markov Chains Open
This monograph is an exposition on an exciting new technique known as spectral independence, which has been instrumental in analyzing the convergence rate of Markov Chain Monte Carlo (MCMC) algorithms. For a high-dimensional distribution d…
View article: Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees
Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees Open
We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more…
View article: Implementations and the independent set polynomial below the Shearer threshold
Implementations and the independent set polynomial below the Shearer threshold Open
The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold is the value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It is known that for λ<−λ⁎(Δ), there are gr…
View article: Beyond the Existential Theory of the Reals
Beyond the Existential Theory of the Reals Open
We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger complete…
View article: Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling
Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling Open
We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution $μ$, an $\varepsilon>0$, and access to sampling oracle(s) for a hidden distribution $π$, the goal in identity testing is to d…
View article: Approximating observables is as hard as counting
Approximating observables is as hard as counting Open
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximati…
View article: Spiraling and Folding: The Topological View
Spiraling and Folding: The Topological View Open
For every $n$, we construct two curves in the plane that intersect at least $n$ times and do not form spirals. The construction is in three stages: we first exhibit closed curves on the torus that do not form double spirals, then arcs on t…
View article: Metastability of the Potts ferromagnet on random regular graphs
Metastability of the Potts ferromagnet on random regular graphs Open
We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in t…
View article: Fast Sampling via Spectral Independence Beyond Bounded-Degree Graphs
Fast Sampling via Spectral Independence Beyond Bounded-Degree Graphs Open
Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs fo…
View article: The Degenerate Crossing Number and Higher-Genus Embeddings
The Degenerate Crossing Number and Higher-Genus Embeddings Open
If a graph embeds in a surface with $k$ crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the…
View article: On mixing of Markov chains: coupling, spectral independence, and entropy factorization
On mixing of Markov chains: coupling, spectral independence, and entropy factorization Open
For general spin systems, we prove that a contractive coupling for an arbitrary local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dyn…
View article: Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region
Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region Open
For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bo…
View article: The Complexity of Approximating the Matching Polynomial in the Complex Plane
The Complexity of Approximating the Matching Polynomial in the Complex Plane Open
We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits …
View article: On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization
On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization Open
For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, ar…
View article: The Swendsen-Wang Dynamics on Trees
The Swendsen-Wang Dynamics on Trees Open
The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of…
View article: Rapid Mixing for Colorings via Spectral Independence
Rapid Mixing for Colorings via Spectral Independence Open
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted indepen…
View article: The complexity of approximating averages on bounded-degree graphs
The complexity of approximating averages on bounded-degree graphs Open
We prove that, unless P=NP, there is no polynomial-time algorithm to approximate within some multiplicative constant the average size of an independent set in graphs of maximum degree 6. This is a special case of a more general result for …
View article: Rapid Mixing for Colorings via Spectral Independence
Rapid Mixing for Colorings via Spectral Independence Open
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted indepen…
View article: The Swendsen-Wang Dynamics on Trees
The Swendsen-Wang Dynamics on Trees Open
The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of…
View article: Understanding Diversity based Pruning of Neural Networks - Statistical Mechanical Analysis.
Understanding Diversity based Pruning of Neural Networks - Statistical Mechanical Analysis. Open
Deep learning architectures with a huge number of parameters are often compressed using pruning techniques to ensure computational efficiency of inference during deployment. Despite multitude of empirical advances, there is no theoretical …
View article: Statistical Mechanical Analysis of Neural Network Pruning
Statistical Mechanical Analysis of Neural Network Pruning Open
Deep learning architectures with a huge number of parameters are often compressed using pruning techniques to ensure computational efficiency of inference during deployment. Despite multitude of empirical advances, there is a lack of theor…
View article: Structure Learning of H-Colorings
Structure Learning of H-Colorings Open
We study the following structure learning problem for H -colorings. For a fixed (and known) constraint graph H with q colors, given access to uniformly random H -colorings of an unknown graph G=(V,E) , how many samples are required to lear…
View article: Hardness of Identity Testing for Restricted Boltzmann Machines and Potts models
Hardness of Identity Testing for Restricted Boltzmann Machines and Potts models Open
We study identity testing for restricted Boltzmann machines (RBMs), and more generally for undirected graphical models. Given sample access to the Gibbs distribution corresponding to an unknown or hidden model $M^*$ and given an explicit m…
View article: Improved Strong Spatial Mixing for Colorings on Trees
Improved Strong Spatial Mixing for Colorings on Trees Open
Strong spatial mixing (SSM) is a form of correlation decay that has played an essential role in the design of approximate counting algorithms for spin systems. A notable example is the algorithm of Weitz (2006) for the hard-core model on w…