Inspired by Hilberg's hypothesis, which states that mutual information between blocks for natural language grows like a power law, we seek for links between power-law growth rate of algorithmic mutual information and of some estimator of the unifilar order, i.e., the number of hidden states in the generating stationary ergodic source in its minimal unifilar hidden Markov representation. We consider an order estimator which returns the smallest order for which the maximum likelihood is larger than a weakly penalized universal probability. This order estimator is intractable and follows the ideas by Merhav, Gutman, and Ziv (1989) and by Ziv and Merhav (1992) but in its exact form seems overlooked despite some nice theoretical properties. In particular, we can prove both strong consistency of this order estimator and an upper bound of algorithmic mutual information in terms of it. Using both results, we show that all (also uncomputable) sources of a finite unifilar order exhibit sub-power-law growth of algorithmic mutual information and of the unifilar order estimator. In contrast, we also exhibit an example of unifilar processes of a countably infinite order, with a deterministic pushdown automaton and an algorithmically random oracle, for which the mentioned two quantities grow as a power law with the same exponent. We also relate our results to natural language research.
Given two events $A$ and $B$, Bayes' law is based on the argument that the probability of $A$ given $B$ is proportional to the probability of $B$ given $A$. When probabilities are interpreted in the Bayesian sense, Bayes' law constitutes a learning algorithm which shows how one can learn from a new observation to improve their belief in a theory that is consistent with that observation. Kolmogorov's notion of algorithmic information, which is based on the theory of algorithms, proposes an objective measure of the amount of information in a finite string about itself and concludes that for any …
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings $x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having $x$ and the complexity profile of the pair and the other one having $y$ and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, \ldots , x_\ell)$ by $\ell$ parties, each one having one component of the tuple and …
In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finite-state dimension (a finite-state version of algorithmic dimension) of a sequence $S \in Σ^\infty$ and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences $S \in Σ^\infty$ and $T \in Σ^\infty$. In this paper, we propose a definition for the lower and upper finite-state mutual dimensions $mdim_{FS}(S:T)$ and $Mdim_{FS}(S:T)$ between two sequences $S \in Σ^\infty$ and $T \in Σ^\infty$ over an alphabet $Σ$. Intuitively, the finite-state dimension of a sequence $S \in …
We rigorously derive a single-letter variational expression for the mutual information of the asymmetric two-groups stochastic block model in the dense graph regime. Existing proofs in the literature are indirect, as they involve mapping the model to a rank-one matrix estimation problem whose mutual information is then determined by a combination of methods (e.g., interpolation, cavity, algorithmic, spatial coupling). In this contribution we provide a self-contained direct method using only the recently introduced adaptive interpolation method.
Markov random fields area popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models. Here we take a more conceptual approach to proving lower bounds on the mutual informati…
Sensors often serve at least two purposes: predicting their input and minimizing dissipated heat. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is difficult. This arises partly from the functional constraints being at cross purposes and partly since quantifying the predictive performance of even in silico sensors can require prohibitively long simulations. To circumvent these difficulties, we develop expressions for the predictive accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to unifi…
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；International Center for Theoretical Physics（国际理论物理中心）；CNRS（法国国家科学研究中心）；PSL Universités（巴黎文理研究大学）；Ecole Normale Supérieure（巴黎高等师范学院）；Sorbonne Universités（索邦大学）；Université Pierre & Marie Curie（皮埃尔与玛丽·居里大学）；CEA（法国原子能和替代能源委员会）；Université Paris-Saclay（巴黎萨克雷大学）
Factorizing low-rank matrices is a problem with many applications in machine learning and statistics, ranging from sparse PCA to community detection and sub-matrix localization. For probabilistic models in the Bayes optimal setting, general expressions for the mutual information have been proposed using powerful heuristic statistical physics computations via the replica and cavity methods, and proven in few specific cases by a variety of methods. Here, we use the spatial coupling methodology developed in the framework of error correcting codes, to rigorously derive the mutual information for t…
We consider a channel with a binary input X being corrupted by a continuous-valued noise that results in a continuous-valued output Y. An optimal binary quantizer is used to quantize the continuous-valued output Y to the final binary output Z to maximize the mutual information I(X; Z). We show that when the ratio of the channel conditional density r(y) = P(Y=y|X=0)/ P(Y =y|X=1) is a strictly increasing/decreasing function of y, then a quantizer having a single threshold can maximize mutual information. Furthermore, we show that an optimal quantizer (possibly with multiple thresholds) is the on…
We formulate an info-clustering paradigm based on a multivariate information measure, called multivariate mutual information, that naturally extends Shannon's mutual information between two random variables to the multivariate case involving more than two random variables. With proper model reductions, we show that the paradigm can be applied to study the human genome and connectome in a more meaningful way than the conventional algorithmic approach. Not only can info-clustering provide justifications and refinements to some existing techniques, but it also inspires new computationally feasibl…
Tighter Bounds for Algorithmic Complexity Estimation Using a Reusable Code-Based Block Decomposition Method
Algorithmic Dynamics Lab, Karolinska Institute & King’s College London（算法动力学实验室，卡罗林斯卡学院与伦敦国王学院）；School of Electrical and Computer Engineering, University of Campinas (UNICAMP)（坎皮纳斯大学电气与计算机工程学院）；Oxford Immune Algorithmics, Oxford University Innovation & London Institute for Healthcare Engineering（牛津免疫算法公司，牛津大学创新部与伦敦医疗工程研究所）；Centre for Logic, Epistemology and the History of Science, University of Campinas (UNICAMP)（坎皮纳斯大学逻辑、认识论与科学史中心）；DEXL, National Laboratory for Scientific Computing (LNCC)（DEXL，巴西国家科学计算实验室）；Departments of Biomedical Computing and Digital Twins, School of Biomedical Engineering and Medical Sciences, King’s Faculty of Life Sciences and Medicine, King’s College London（伦敦国王大学生物医学与医学科学学院生物医学计算与数字孪生系）；King’s Institute for Artificial Intelligence, King’s College London（伦敦国王学院人工智能研究所）
University of Cambridge（剑桥大学）；Rutgers University（罗格斯大学）；Imperial College London（伦敦帝国理工学院）；Alan Turing Institute（艾伦·图灵研究所）
The task of infomin learning aims to learn a representation with high utility while being uninformative about a specified target, with the latter achieved by minimising the mutual information between the representation and the target. It has broad applications, ranging from training fair prediction models against protected attributes, to unsupervised learning with disentangled representations. Recent works on infomin learning mainly use adversarial training, which involves training a neural network to estimate mutual information or its proxy and thus is slow and difficult to optimise. Drawing …
We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections. A few examples where this problem is relevant are compressed sensing, sparse superposition codes, and code division multiple access. There has been a number of works considering the mutual information for this problem using the replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-Toninelli type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relev…
Previous work has shown that perturbation analysis in software space can produce candidate computable generative models and uncover possible causal properties from the finite description of an object or system quantifying the algorithmic contribution of each of its elements relative to the whole. One of the challenges for defining emergence is that one observer's prior knowledge may cause a phenomenon to present itself to such observer as emergent while for another as reducible. By formalising the act of observing as mutual perturbations between dynamical systems, we demonstrate that emergence…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Arizona State University（亚利桑那州立大学）；University of Arkansas（阿肯色大学）
Algorithmic fairness is becoming increasingly important in data mining and machine learning. Among others, a foundational notation is group fairness. The vast majority of the existing works on group fairness, with a few exceptions, primarily focus on debiasing with respect to a single sensitive attribute, despite the fact that the co-existence of multiple sensitive attributes (e.g., gender, race, marital status, etc.) in the real-world is commonplace. As such, methods that can ensure a fair learning outcome with respect to all sensitive attributes of concern simultaneously need to be developed…
Recent work has established that the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020) is expressive enough to capture generalization guarantees in terms of algorithmic stability, VC dimension, and related complexity measures for conventional learning (Harutyunyan et al., 2021, Haghifam et al., 2021). Hence, it provides a unified method for establishing generalization bounds. In meta learning, there has so far been a divide between information-theoretic results and results from classical learning theory. In this work, we take a first step toward bridging this div…
