scholarly journals Faster quantum mixing for slowly evolving sequences of Markov chains

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 105 ◽  
Author(s):  
Davide Orsucci ◽  
Hans J. Briegel ◽  
Vedran Dunjko
Keyword(s):  
Machine Learning ◽  
Markov Chain ◽  
Markov Chains ◽  
Quantum Algorithms ◽  
Mixing Time ◽  
Classical Case ◽  
Special Cases ◽  
Run Time ◽  
Time Required ◽  
The Cost

Markov chain methods are remarkably successful in computational physics, machine learning, and combinatorial optimization. The cost of such methods often reduces to the mixing time, i.e., the time required to reach the steady state of the Markov chain, which scales asδ−1, the inverse of the spectral gap. It has long been conjectured that quantum computers offer nearly generic quadratic improvements for mixing problems. However, except in special cases, quantum algorithms achieve a run-time ofO(δ−1N), which introduces a costly dependence on the Markov chain sizeN,not present in the classical case. Here, we re-address the problem of mixing of Markov chains when these form a slowly evolving sequence. This setting is akin to the simulated annealing setting and is commonly encountered in physics, material sciences and machine learning. We provide a quantum memory-efficient algorithm with a run-time ofO(δ−1N4), neglecting logarithmic terms, which is an important improvement for large state spaces. Moreover, our algorithms output quantum encodings of distributions, which has advantages over classical outputs. Finally, we discuss the run-time bounds of mixing algorithms and show that, under certain assumptions, our algorithms are optimal.

scholarly journals THE DISTRIBUTION OF MIXING TIMES IN MARKOV CHAINS

2013 ◽  
Vol 30 (01) ◽  
pp. 1250045 ◽  
Author(s):  
JEFFREY J. HUNTER
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Linear Algebra ◽  
Mixing Time ◽  
Probability Generating Function ◽  
First Passage Times ◽  
Mixing Times ◽  
First Passage ◽  
Special Cases ◽  
Passage Times

The distribution of the "mixing time" or the "time to stationarity" in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time, starting in state i, are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [Hunter, JJ (2006). Mixing times with applications to perturbed Markov chains. Linear Algebra and Its Applications, 417, 108–123] and the variance of the times to mixing, [Hunter, JJ (2008). Variances of first passage times in a Markov chain with applications to mixing times. Linear Algebra and Its Applications, 429, 1135–1162]. Some new results for the distribution of the recurrence and the first passage times in a general irreducible three-state Markov chain are also presented.

scholarly journals Smooth input preparation for quantum and quantum-inspired machine learning

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Zhikuan Zhao ◽  
Jack K. Fitzsimons ◽  
Patrick Rebentrost ◽  
Vedran Dunjko ◽  
Joseph F. Fitzsimons
Keyword(s):  
Machine Learning ◽  
Quantum Algorithms ◽  
Machine Learning Algorithms ◽  
Low Rank ◽  
Smoothed Analysis ◽  
Machine Learning Applications ◽  
Data Points ◽  
Input Model ◽  
The Cost ◽  
Input Perturbation

AbstractMachine learning has recently emerged as a fruitful area for finding potential quantum computational advantage. Many of the quantum-enhanced machine learning algorithms critically hinge upon the ability to efficiently produce states proportional to high-dimensional data points stored in a quantum accessible memory. Even given query access to exponentially many entries stored in a database, the construction of which is considered a one-off overhead, it has been argued that the cost of preparing such amplitude-encoded states may offset any exponential quantum advantage. Here we prove using smoothed analysis that if the data analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries. This criterion is typically satisfied in realistic machine learning applications, where input data is subjective to moderate noise. Our results are equally applicable to the recent seminal progress in quantum-inspired algorithms, where specially constructed databases suffice for polylogarithmic classical algorithm in low-rank cases. The consequence of our finding is that for the purpose of practical machine learning, polylogarithmic processing time is possible under a general and flexible input model with quantum algorithms or quantum-inspired classical algorithms in the low-rank cases.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

scholarly journals Run-Time and Compiler Support for Programming in Adaptive Parallel Environments

1997 ◽  
Vol 6 (2) ◽  
pp. 215-227 ◽  
Author(s):  
Guy Edjlali ◽  
Gagan Guyagrawal ◽  
Alan Sussman ◽  
Jim Humphries ◽  
Joel Saltz
Keyword(s):  
Parallel Programming ◽  
High Performance ◽  
Navier Stokes ◽  
Programming Environments ◽  
Data Parallel ◽  
Adaptive Environment ◽  
Run Time ◽  
Time Required ◽  
The Cost ◽  
Performance Results

For better utilization of computing resources, it is important to consider parallel programming environments in which the number of available processors varies at run-time. In this article, we discuss run-time support for data-parallel programming in such an adaptive environment. Executing programs in an adaptive environment requires redistributing data when the number of processors changes, and also requires determining new loop bounds and communication patterns for the new set of processors. We have developed a run-time library to provide this support. We discuss how the run-time library can be used by compilers of high-performance Fortran (HPF)-like languages to generate code for an adaptive environment. We present performance results for a Navier-Stokes solver and a multigrid template run on a network of workstations and an IBM SP-2. Our experiments show that if the number of processors is not varied frequently, the cost of data redistribution is not significant compared to the time required for the actual computation. Overall, our work establishes the feasibility of compiling HPF for a network of nondedicated workstations, which are likely to be an important resource for parallel programming in the future.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (01) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

scholarly journals Scheduling of Preventive Maintenance in Healthcare Buildings Using Markov Chain

2020 ◽  
Vol 10 (15) ◽  
pp. 5263
Author(s):  
Jaime González-Domínguez ◽  
Gonzalo Sánchez-Barroso ◽  
Justo García-Sanz-Calcedo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Preventive Maintenance ◽  
Operating Costs ◽  
Use Of Resources ◽  
Overall Efficiency ◽  
Maintenance Policies ◽  
Reduce Emissions ◽  
The Cost ◽  
Specific Degradation

The optimization of maintenance in healthcare buildings reduces operating costs and contributes towards increasing the sustainability of the healthcare system. This paper proposes a tool to schedule preventive maintenance for healthcare centers using Markov chains. To this end, the authors analyzed 25 healthcare centers belonging to the three Healthcare Districts of Spain and built between 1985 and 2005. Markov chains proved useful in choosing the most suitable maintenance policies for each healthcare building without exceeding a specific degradation boundary, which enabled achieving an ideal maintenance frequency and reduced the use of resources. Markov chains have also proven useful in optimizing the periodicity of routine maintenance tasks, ensuring a suitable level of maintenance according to the frequency of the failures and reducing the cost and carbon footprint. The healthcare centers observed during the study managed to save more than 700 km of journeys, reduce emissions in its operations as a whole by 174.3 kg of CO2 per month and increase the overall efficiency of maintenance operations by 15%. This approach, therefore, renders it advisable to plan the maintenance of healthcare buildings.

scholarly journals A Comparative Analysis of Active Learning for Biomedical Text Mining

2021 ◽  
Vol 4 (1) ◽  
pp. 23
Author(s):  
Usman Naseem ◽  
Matloob Khushi ◽  
Shah Khalid Khan ◽  
Kamran Shaukat ◽  
Mohammad Ali Moni
Keyword(s):  
Machine Learning ◽  
Active Learning ◽  
De Novo ◽  
Comparative Investigation ◽  
Free Text ◽  
Machine Learning Applications ◽  
Text Information ◽  
Pathology Reports ◽  
Time Required ◽  
The Cost

An enormous amount of clinical free-text information, such as pathology reports, progress reports, clinical notes and discharge summaries have been collected at hospitals and medical care clinics. These data provide an opportunity of developing many useful machine learning applications if the data could be transferred into a learn-able structure with appropriate labels for supervised learning. The annotation of this data has to be performed by qualified clinical experts, hence, limiting the use of this data due to the high cost of annotation. An underutilised technique of machine learning that can label new data called active learning (AL) is a promising candidate to address the high cost of the label the data. AL has been successfully applied to labelling speech recognition and text classification, however, there is a lack of literature investigating its use for clinical purposes. We performed a comparative investigation of various AL techniques using ML and deep learning (DL)-based strategies on three unique biomedical datasets. We investigated random sampling (RS), least confidence (LC), informative diversity and density (IDD), margin and maximum representativeness-diversity (MRD) AL query strategies. Our experiments show that AL has the potential to significantly reducing the cost of manual labelling. Furthermore, pre-labelling performed using AL expediates the labelling process by reducing the time required for labelling.

scholarly journals Quantum fast-forwarding: Markov chains and graph property testing

2019 ◽  
Vol 19 (3&4) ◽  
pp. 181-213 ◽  
Author(s):  
Simon Apers ◽  
Alain Scarlet
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Quantum State ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Walk ◽  
Transient Behavior ◽  
Quantum Walks ◽  
Learning Context

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

scholarly journals On Sampling Colorings of Bipartite Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
R. Balasubramanian ◽  
C.R. Subramanian
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Mixing Time ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Negative Results ◽  
Bipartite Subgraph ◽  
Single Transition ◽  
International Audience ◽  
Complete Bipartite Subgraph

International audience We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.

scholarly journals Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

2017 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Stefan Felsner ◽  
Daniel Heldt
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Distributive Lattice ◽  
Maximum Degree ◽  
Lattice Structure ◽  
Mixing Time ◽  
Plane Graph ◽  
Negative Results ◽  
Slow Mixing ◽  
Constant Maximum

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.

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