A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least αα, for some α∈(0,1)α∈(0,1). We also give general bounds on the total variation distance of a reversible chain at time tt in terms of the probability that some “worst” set of stationary measure at least αα was not hit by time tt. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞∞.
This work was greatly motivated by the aforementioned results. It is natural to ask whether ) could be further refined so that the cutoff phenomenon could be characterized in terms of concentration of the hitting times of a sequence of sets @math which attain the maximum in the definition of @math (starting from the worst initial states). Corollary 1.5 in @cite_8 asserts that this is indeed the case in the transitive setup. More generally, Theorem 2 in @cite_8 asserts that this is indeed the case for any fixed sequence of initial states @math if one replaces @math and @math by @math and @math (i.e. when the hitting times and the mixing times are defined only w.r.t. these starting states). Alas, Proposition 1.6 in @cite_8 asserts that in general cutoff could not be characterized in this manner.
@cite_8
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Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner. Finally, we also prove that there exists an absolute constant @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain. A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given. This technique is used to prove (i) A tight bound of @math for the two-dimensional torus; (ii) A tight bound of @math for trees with maximum degree @math ; (iii) Tight bounds of @math for rapidly mixing walks on vertex transitive graphs, where @math denotes the maximum expected hitting time between vertices.In addition to these new results, the technique allows several known lower bounds on cover times to be systematically proved, often in a much simpler way.Finally, a different technique is used to prove an @math lower bound on the cover time, where @math is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [... We consider the simple exclusion process with @math particles on a segment of length @math performing random walks with transition @math to the right and @math to the left. We focus on the case where the asymmetry in the jump rates @math vanishes in the limit when @math and @math tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than @math . We show that in the former case ( @math ), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant @math ) case studied in [LL18], while the latter case ( @math ) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [Lac16b]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order @math . A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least αα, for some α∈(0,1)α∈(0,1). We also give general bounds on the total variation distance of a reversible chain at time tt in terms of the probability that some “worst” set of stationary measure at least αα was not hit by time tt. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞∞. Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. A family @math of graphs on a fixed set of @math vertices is called triangle-intersecting if for any @math , the intersection @math contains a triangle. More generally, for a fixed graph @math , a family @math is @math -intersecting if the intersection of any two graphs in @math contains a sub-graph isomorphic to @math . In [D. Ellis, Y. Filmus, and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. 14 (2012), pp. 841--885], Ellis, Filmus and Friedgut proved a 36-year old conjecture of Simonovits and Sos stating that the maximal size of a triangle-intersecting family is @math . Furthermore, they proved a @math -biased generalization, stating that for any @math , we have @math , where @math is the probability that the random graph @math belongs to @math . In the same paper, conjectured that the assertion of their biased theorem holds also for @math , and more generally, that for any non- @math -colorable graph @math and any @math -intersecting family @math , we have @math for all @math . In this note we construct, for any fixed @math and any @math , an @math -intersecting family @math of graphs such that @math , where @math depends only on @math and @math , thus disproving both conjectures. The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree @math which satisfies @math , where @math is the sum of the degrees. The mixing time bound obtained is only a factor @math larger than that established in the regular case, where @math is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter @math and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by @math , where @math is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of @math larger than that established in the regular case. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. We study the cover time of multiple random walks. Given a graph G of n vertices, assume that k independent random walks start from the same vertex. The parameter of interest is the speed-up defined as the ratio between the cover time of one and the cover time of k random walks. Recently developed several bounds that are based on the quotient between the cover time and maximum hitting times. Their technique gives a speed-up of *** (k ) on many graphs, however, for many graph classes, k has to be bounded by @math . They also conjectured that, for any 1 ≤ k ≤ n , the speed-up is at most @math on any graph. As our main results, we prove the following: We present a new lower bound on the speed-up that depends on the mixing-time. It gives a speed-up of *** (k ) on many graphs, even if k is as large as n . We prove that the speed-up is @math on any graph. Under rather mild conditions, we can also improve this bound to @math , matching exactly the conjecture of We find the correct order of the speed-up for any value of 1 ≤ k ≤ n on hypercubes, random graphs and expanders. For d -dimensional torus graphs (d > 2), our bounds are tight up to a factor of @math . Our findings also reveal a surprisingly sharp dichotomy on several graphs (including d -dim. torus and hypercubes): up to a certain threshold the speed-up is k , while there is no additional speed-up above the threshold. Let @math . We show that that the mixing time of a continuous-time Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of the state space with stationary measure @math . Suitably modified results hold in discrete time and or without the reversibility assumption. The key technical tool in the proof is the construction of random set @math such that the hitting time of @math is a light-tailed stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi. Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. Deep neural networks (DNNs) have begun to have a pervasive impact on various applications of machine learning. However, the problem of finding an optimal DNN architecture for large applications is challenging. Common approaches go for deeper and larger DNN architectures but may incur substantial redundancy. To address these problems, we introduce a network growth algorithm that complements network pruning to learn both weights and compact DNN architectures during training. We propose a DNN synthesis tool (NeST) that combines both methods to automate the generation of compact and accurate DNNs. NeST starts with a randomly initialized sparse network called the seed architecture. It iteratively tunes the architecture with gradient-based growth and magnitude-based pruning of neurons and connections. Our experimental results show that NeST yields accurate, yet very compact DNNs, with a wide range of seed architecture selection. For the LeNet-300-100 (LeNet-5) architecture, we reduce network parameters by @math 70.2× ( @math 74.3×) and floating-point operations (FLOPs) by @math 79.4× ( @math 43.7×). For the AlexNet, VGG-16, and ResNet-50 architectures, we reduce network parameters (FLOPs) by @math 15.7× ( @math 4.6×), @math 33.2× ( @math 8.9×), and @math 4.1× ( @math 2.1×) respectively. NeST's grow-and-prune paradigm delivers significant additional parameter and FLOPs reduction relative to pruning-only methods. We study the localization problem in the ring: a collection of @math n anonymous mobile robots are deployed in a continuous ring of perimeter one. All robots start moving at the same time with arbitrary velocities, starting in clockwise or counterclockwise direction around the ring. The robots bounce against each other according to the principles of conservation of energy and momentum. The task of each robot is to find out, in finite time, the initial position and the initial velocity of every deployed robot. The only way that robots perceive the information about the environment is by colliding with their neighbors; robots have no control of their walks or velocities moreover any type of communication among them is not possible. The configuration of initial positions of robots and their speeds is considered feasible, if there is a finite time, after which every robot starting at this configuration knows initial positions and velocities of all other robots. It was conjectured in (2012) that if the principles of conservation of energy and momentum were assumed and the robots had arbitrary velocities, the localization problem might be solvable. We prove that this conjecture is false. We show that if @math v0,v1,?,vn-1 are the velocities of a given robot configuration @math S, then @math S is feasible if and only if @math vi?v¯ for all @math 0≤i≤n-1, where @math v¯=v0+?+vn-1n. To figure out the initial positions of all robots no more than @math 2min0≤i≤n-1|vi-v¯| time is required. Given a finite graph @math , a vertex of the lamplighter graph @math consists of a zero-one labeling of the vertices of @math , and a marked vertex of @math . For transitive @math we show that, up to constants, the relaxation time for simple random walk in @math is the maximal hitting time for simple random walk in @math , while the mixing time in total variation on @math is the expected cover time on @math . The mixing time in the uniform metric on @math admits a sharp threshold, and equals @math multiplied by the relaxation time on @math , up to a factor of @math . For @math , the lamplighter group over the discrete two dimensional torus, the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . For @math when @math , the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . In particular, these three quantities are of different orders of magnitude. High energy efficiency and low memory footprint are the key requirements for the deployment of deep learning based analytics on low-power microcontrollers. Here we present work-in-progress results with @math -bit Quantized Neural Networks (QNNs) deployed on a commercial Cortex-M7 class microcontroller by means of an extension to the ARM CMSIS-NN library. We show that i) for @math and @math low memory footprint QNNs can be deployed with an energy overhead of 30 and 36 respectively against the 8-bit CMSIS-NN due to the lack of quantization support in the ISA; ii) for @math native instructions can be used, yielding an energy and latency reduction of ∼3.8× with respect to CMSIS-NN. Our initial results suggest that a small set of QNN-related specialized instructions could improve performance by as much as 7.5× for @math , 13.6× for @math and 6.5× for binary NNs. A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. A set family F is said to be t-intersecting if any two sets in F share at least t elements. The Complete Intersection Theorem of Ahlswede and Khachatrian (1997) determines the maximal size f(n,k,t) of a t-intersecting family of k-element subsets of 1,2,...,n , and gives a full characterisation of the extremal families. In this paper, we prove the following stability version' of the theorem: if k n is bounded away from 0 and 1 2, and F is a t-intersecting family of k-element subsets of 1,2,...,n such that @math , then there exists an extremal family G such that @math . For fixed t, this assertion is tight up to a constant factor. This proves a conjecture of Friedgut from 2008. Our proof combines classical shifting arguments with a bootstrapping' method based upon an isoperimetric argument. Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on @math -intersecting families of @math -element subsets of @math (for @math ), and to show that, somewhat surprisingly, all these results hold when the intersection requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sos conjecture, and various EKR-type results on @math -wise (cross)- @math -intersecting families. We present techniques to parallelize membership tests for Deterministic Finite Automata (DFAs). Our method searches arbitrary regular expressions by matching multiple bytes in parallel using speculation. We partition the input string into chunks, match chunks in parallel, and combine the matching results. Our parallel matching algorithm exploits structural DFA properties to minimize the speculative overhead. Unlike previous approaches, our speculation is failure-free, i.e., (1) sequential semantics are maintained, and (2) speed-downs are avoided altogether. On architectures with a SIMD gather-operation for indexed memory loads, our matching operation is fully vectorized. The proposed load-balancing scheme uses an off-line profiling step to determine the matching capacity of each participating processor. Based on matching capacities, DFA matches are load-balanced on inhomogeneous parallel architectures such as cloud computing environments. We evaluated our speculative DFA membership test for a representative set of benchmarks from the Perl-compatible Regular Expression (PCRE) library and the PROSITE protein database. Evaluation was conducted on a 4 CPU (40 cores) shared-memory node of the Intel Academic Program Manycore Testing Lab (Intel MTL), on the Intel AVX2 SDE simulator for 8-way fully vectorized SIMD execution, and on a 20-node (288 cores) cluster on the Amazon EC2 computing cloud. Obtained speedups are on the order of @math O 1 + | P | - 1 | Q | · ? , where @math | P | denotes the number of processors or SIMD units, @math | Q | denotes the number of DFA states, and @math 0 < ? ≤ 1 represents a statically computed DFA property. For all observed cases, we found that @math 0.02 < ? < 0.47 . Actual speedups range from 2.3 @math × to 38.8 @math × for up to 512 DFA states for PCRE, and between 1.3 @math × and 19.9 @math × for up to 1,288 DFA states for PROSITE on a 40-core MTL node. Speedups on the EC2 computing cloud range from 5.0 @math × to 65.8 @math × for PCRE, and from 5.0 @math × to 138.5 @math × for PROSITE. Speedups of our C-based DFA matcher over the Perl-based ScanProsite scan tool range from 559.3 @math × to 15079.7 @math × on a 40-core MTL node. We show the scalability of our approach for input-sizes of up to 10 GB.