415 research outputs found
Quantum Algorithm Implementations for Beginners
As quantum computers become available to the general public, the need has
arisen to train a cohort of quantum programmers, many of whom have been
developing classical computer programs for most of their careers. While
currently available quantum computers have less than 100 qubits, quantum
computing hardware is widely expected to grow in terms of qubit count, quality,
and connectivity. This review aims to explain the principles of quantum
programming, which are quite different from classical programming, with
straightforward algebra that makes understanding of the underlying fascinating
quantum mechanical principles optional. We give an introduction to quantum
computing algorithms and their implementation on real quantum hardware. We
survey 20 different quantum algorithms, attempting to describe each in a
succinct and self-contained fashion. We show how these algorithms can be
implemented on IBM's quantum computer, and in each case, we discuss the results
of the implementation with respect to differences between the simulator and the
actual hardware runs. This article introduces computer scientists, physicists,
and engineers to quantum algorithms and provides a blueprint for their
implementations
Market state discovery
We explore the concept of financial market state discovery by assessing the robustness of two unsupervised machine learning algorithms: Inverse Covariance Clustering (ICC) and Agglomerative Super Paramagnetic Clustering (ASPC). The assessment is carried out by: simulating market datasets varying in complexity; implementing ICC and ASPC to estimate the underlying states (using only simulated log-returns as inputs); and measuring the algorithms' ability to recover the underlying states, using the Adjusted Rand Index (ARI) as a performance metric. Experiments revealed that ASPC is a more robust and better performing algorithm than ICC. ICC is able to produce competitive results in 2-state markets; however, ICC's primary disadvantage is its inability to maintain strong performance in 3, 4 and 5-state markets. For example, ASPC produced ARI numbers that were up to 800% superior to ICC in 5-state markets. Furthermore, ASPC does not rely on the art of selecting good hyper-parameters such as, the number of states a priori. ICC's utility as a market state discovery algorithm is limited
2D growth processes: SLE and Loewner chains
This review provides an introduction to two dimensional growth processes.
Although it covers a variety processes such as diffusion limited aggregation,
it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner
evolutions (SLE) which are Markov processes describing interfaces in 2D
critical systems. It starts with an informal discussion, using numerical
simulations, of various examples of 2D growth processes and their connections
with statistical mechanics. SLE is then introduced and Schramm's argument
mapping conformally invariant interfaces to SLE is explained. A substantial
part of the review is devoted to reveal the deep connections between
statistical mechanics and processes, and more specifically to the present
context, between 2D critical systems and SLE. Some of the SLE remarkable
properties are explained, as well as the tools for computing with SLE. This
review has been written with the aim of filling the gap between the
mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172
pages, low quality figures, better quality figures upon request to the
authors, comments welcom
A network approach to topic models
One of the main computational and scientific challenges in the modern age is
to extract useful information from unstructured texts. Topic models are one
popular machine-learning approach which infers the latent topical structure of
a collection of documents. Despite their success --- in particular of its most
widely used variant called Latent Dirichlet Allocation (LDA) --- and numerous
applications in sociology, history, and linguistics, topic models are known to
suffer from severe conceptual and practical problems, e.g. a lack of
justification for the Bayesian priors, discrepancies with statistical
properties of real texts, and the inability to properly choose the number of
topics. Here we obtain a fresh view on the problem of identifying topical
structures by relating it to the problem of finding communities in complex
networks. This is achieved by representing text corpora as bipartite networks
of documents and words. By adapting existing community-detection methods --
using a stochastic block model (SBM) with non-parametric priors -- we obtain a
more versatile and principled framework for topic modeling (e.g., it
automatically detects the number of topics and hierarchically clusters both the
words and documents). The analysis of artificial and real corpora demonstrates
that our SBM approach leads to better topic models than LDA in terms of
statistical model selection. More importantly, our work shows how to formally
relate methods from community detection and topic modeling, opening the
possibility of cross-fertilization between these two fields.Comment: 22 pages, 10 figures, code available at https://topsbm.github.io
Unsupervised inference methods for protein sequence data
L'abstract Ăš presente nell'allegato / the abstract is in the attachmen
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Modernizing Markov Chains Monte Carlo for Scientific and Bayesian Modeling
The advent of probabilistic programming languages has galvanized scientists to write increasingly diverse models to analyze data. Probabilistic models use a joint distribution over observed and latent variables to describe at once elaborate scientific theories, non-trivial measurement procedures, information from previous studies, and more. To effectively deploy these models in a data analysis, we need inference procedures which are reliable, flexible, and fast. In a Bayesian analysis, inference boils down to estimating the expectation values and quantiles of the unnormalized posterior distribution. This estimation problem also arises in the study of non-Bayesian probabilistic models, a prominent example being the Ising model of Statistical Physics.
Markov chains Monte Carlo (MCMC) algorithms provide a general-purpose sampling method which can be used to construct sample estimators of moments and quantiles. Despite MCMCâs compelling theory and empirical success, many models continue to frustrate MCMC, as well as other inference strategies, effectively limiting our ability to use these models in a data analysis. These challenges motivate new developments in MCMC. The term âmodernizeâ in the title refers to the deployment of methods which have revolutionized Computational Statistics and Machine Learning in the past decade, including: (i) hardware accelerators to support massive parallelization, (ii) approximate inference based on tractable densities, (iii) high-performance automatic differentiation and (iv) continuous relaxations of discrete systems.
The growing availability of hardware accelerators such as GPUs has in the past years motivated a general MCMC strategy, whereby we run many chains in parallel with a short sampling phase, rather than a few chains with a long sampling phase. Unfortunately existing convergence diagnostics are not designed for the âmany short chainsâ regime. This is notably the case of the popular R statistics which claims convergence only if the effective sample size per chain is large. We present the nested R, denoted nR, a generalization of R which does not conflate short chains and poor mixing, and offers a useful diagnostic provided we run enough chains and meet certain initialization conditions. Combined with nR the short chain regime presents us with the opportunity to identify optimal lengths for the warmup and sampling phases, as well as the optimal number of chains; tuning parameters of MCMC which are otherwise chosen using heuristics or trial-and-error.
We next focus on semi-specialized algorithms for latent Gaussian models, arguably the most widely used of class of hierarchical models. It is well understood that MCMC often struggles with the geometry of the posterior distribution generated by these models. Using a Laplace approximation, we marginalize out the latent Gaussian variables and then integrate the remaining parameters with Hamiltonian Monte Carlo (HMC), a gradient-based MCMC. This approach combines MCMC and a distributional approximation, and offers a useful alternative to pure MCMC or pure approximation methods such as Variational Inference. We compare the three paradigms across a range of general linear models, which admit a sophisticated prior, i.e. a Gaussian process and a Horseshoe prior. To implement our scheme efficiently, we derive a novel automatic differentiation method called the adjoint-differentiated Laplace approximation. This differentiation algorithm propagates the minimal information needed to construct the gradient of the approximate marginal likelihood, and yields a scalable differentiation method that is orders of magnitude faster than state of the art differentiation for high-dimensional hyperparameters. We next discuss the application of our algorithm to models with an unconventional likelihood, going beyond the classical setting of general linear models. This necessitates a non-trivial generalization of the adjoint-differentiated Laplace approximation, which we implement using higher-order adjoint methods. The generalization works out to be both more general and more efficient. We apply the resulting method to an unconventional latent Gaussian model, identifying promising features and highlighting persistent challenges.
The final chapter of this dissertation focuses on a specific but rich problem: the Ising model of Statistical Physics, and its generalization as the Potts and Spin Glass models. These models are challenging because they are discrete, precluding the immediate use of gradient-based algorithms, and exhibit multiple modes, notably at cold temperatures. We propose a new class of MCMC algorithms to draw samples from Potts models by augmenting the target space with a carefully constructed auxiliary Gaussian variable. In contrast to existing methods of a similar flavor, our algorithm can take advantage of the low-rank structure of the coupling matrix and scales linearly with the number of states in a Potts model. The method is applied to a broad range of coupling and temperature regimes and compared to several sampling methods, allowing us to paint a nuanced algorithmic landscape
Global and Local Information in Clustering Labeled Block Models
The stochastic block model is a classical cluster-exhibiting random graph
model that has been widely studied in statistics, physics and computer science.
In its simplest form, the model is a random graph with two equal-sized
clusters, with intra-cluster edge probability p, and inter-cluster edge
probability q. We focus on the sparse case, i.e., p, q = O(1/n), which is
practically more relevant and also mathematically more challenging. A
conjecture of Decelle, Krzakala, Moore and Zdeborova, based on ideas from
statistical physics, predicted a specific threshold for clustering. The
negative direction of the conjecture was proved by Mossel, Neeman and Sly
(2012), and more recently the positive direction was proven independently by
Massoulie and Mossel, Neeman, and Sly.
In many real network clustering problems, nodes contain information as well.
We study the interplay between node and network information in clustering by
studying a labeled block model, where in addition to the edge information, the
true cluster labels of a small fraction of the nodes are revealed. In the case
of two clusters, we show that below the threshold, a small amount of node
information does not affect recovery. On the other hand, we show that for any
small amount of information efficient local clustering is achievable as long as
the number of clusters is sufficiently large (as a function of the amount of
revealed information).Comment: 24 pages, 2 figures. A short abstract describing these results will
appear in proceedings of RANDOM 201
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