Dynamical quantum ergodicity from energy level statistics

Ergodic theory provides a rigorous mathematical description of classical dynamical systems including a formal definition of the ergodic hierarchy. Closely related to this hierarchy is a less-known notion of cyclic approximate periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai, Ergodic theory (Springer-Verlag New York, 1982)], which maps any "ergodic" dynamical system to a cyclic permutation on a circle and arguably represents the most elementary notion of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, and this generalization is proposed here as the basic rigorous definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports an initial basis vector to all other basis vectors one by one, while maintaining a sufficiently large overlap between the time-evolved initial state and a given basis state. It is proven that the basis, maximizing the overlap over all cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates. This relates quantum cyclic ergodicity to level statistics. We then show that the near-universal Wigner-Dyson level statistics implies quantum cyclic ergodicity, but the reverse is not necessarily true. For the latter, we study irrational flows on a 2D torus and prove that both the classical and quantum flows are cyclic ergodic. However, the corresponding level statistics is neither Wigner-Dyson nor Poisson. Finally, we use the cyclic construction to motivate a quantum ergodic hierarchy of operators and argue that under the additional assumption of Poincare recurrences, cyclic ergodicity is a necessary condition for such operators to satisfy the eigenstate thermalization hypothesis. This work provides a general framework for transplanting some rigorous concepts of ergodic theory to quantum dynamical systems.Comment: 42+11 pages, 9+1 figures; v2: updated definition of aperiodicity, analytical results for tori, improved presentation and some new figure

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.Comment: 23 pages, 3 figures, 8 table

On the Expressivity of Persistent Homology in Graph Learning

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks

The inscrutability of reference

The metaphysics of representation poses questions such as: in virtue of what does a sentence, picture, or mental state represent that the world is a certain way? In the first instance, I have focused on the semantic properties of language: for example, what is it for a name such as ‘London’ to refer to something? Interpretationism concerning what it is for linguistic expressions to have meaning, says that constitutively, semantic facts are fixed by best semantic theory. As here developed, it promises to give a reductive, universal and non-revisionary account of the nature of linguistic representation. Interpretationism in general, however, is threatened by severe internal tension, due to arguments for radical inscrutability. These contend that, given the interpretationist setting, there can be no fact of the matter what object an individual word refers to: for example, that there is no fact of the matter as to whether “London” refers to London or to Sydney. A series of challenges emerge, forming the basis for this thesis. 1. What sort of properties is the interpretationist trying to reduce, and what kind of reductive story is she offering? 2. How are inscrutability theses best formulated? Are arguments for inscrutability effective in their own terms? What kinds of inscrutability arise? 3. Is endorsing radical inscrutability a stable position? 4. Are there theoretical virtues—such as simplicity—that can be appealed to in discrediting the rival (empirically equivalent) theories that underpin inscrutability arguments? In addressing these questions, I concentrate on diagnosing the source of inscrutability, mapping the space of ways of resisting the arguments for radical inscrutability, and examining the challenges faced in developing a principled account of linguistic content that avoids radical inscrutability. The effect is not to close down the original puzzles, but rather to sharpen them into a set of new and deeper challenges

Flexible non-parametric tests of sample exchangeability and feature independence

In scientific studies involving analyses of multivariate data, two questions often arise for the researcher. First, is the sample exchangeable, meaning that the joint distribution of the sample is invariant to the ordering of the units? Second, are the features independent of one another, or can the features be grouped so that the groups are mutually independent? We propose a non-parametric approach that addresses these two questions. Our approach is conceptually simple, yet fast and flexible. It controls the Type I error across realistic scenarios, and handles data of arbitrary dimensions by leveraging large-sample asymptotics. In the exchangeability detection setting, through extensive simulations and a comparison against unsupervised tests of stratification based on random matrix theory, we find that our approach compares favorably in various scenarios of interest. We apply our method to problems in population and statistical genetics, including stratification detection and linkage disequilibrium splitting. We also consider other application domains, applying our approach to post-clustering single-cell chromatin accessibility data and World Values Survey data, where we show how users can partition features into independent groups, which helps generate new scientific hypotheses about the features.Comment: Main Text: 25 pages Supplementary Material: 39 page

Natural Selection of Immune and Metabolic Genes Associated with Health in Two Lowland Bolivian Populations

A growing body of work has addressed human adaptations to diverse environments using genomic data, but few studies have connected putatively selected alleles to phenotypes, much less among underrepresented populations such as Amerindians. Studies of natural selection and genotype–phenotype relationships in underrepresented populations hold potential to uncover previously undescribed loci underlying evolutionarily and biomedically relevant traits. Here, we worked with the Tsimane and the Moseten, two Amerindian populations inhabiting the Bolivian lowlands. We focused most intensively on the Tsimane, because long-term anthropological work with this group has shown that they have a high burden of both macro and microparasites, as well as minimal cardiometabolic disease or dementia. We therefore generated genome-wide genotype data for Tsimane individuals to study natural selection, and paired this with blood mRNA-seq as well as cardiometabolic and immune biomarker data generated from a larger sample that included both populations. In the Tsimane, we identified 21 regions that are candidates for selective sweeps, as well as 5 immune traits that show evidence for polygenic selection (e.g., C-reactive protein levels and the response to coronaviruses). Genes overlapping candidate regions were strongly enriched for known involvement in immune-related traits, such as abundance of lymphocytes and eosinophils. Importantly, we were also able to draw on extensive phenotype information for the Tsimane and Moseten and link five regions (containing PSD4, MUC21 and MUC22, TOX2, ANXA6, and ABCA1) with biomarkers of immune and metabolic function. Together, our work highlights the utility of pairing evolutionary analyses with anthropological and biomedical data to gain insight into the genetic basis of health-related traits

A new class of neural architectures to model episodic memory : computational studies of distal reward learning

A computational cognitive neuroscience model is proposed, which models episodic memory based on the mammalian brain. A computational neural architecture instantiates the proposed model and is tested on a particular task of distal reward learning. Categorical Neural Semantic Theory informs the architecture design. To experiment upon the computational brain model, embodiment and an environment in which the embodiment exists are simulated. This simulated environment realizes the Morris Water Maze task, a well established biological experimental test of distal reward learning. The embodied neural architecture is treated as a virtual rat and the environment it acts in as a virtual water tank. Performance levels of the neural architectures are evaluated through analysis of embodied behavior in the distal reward learning task. Comparison is made to biological rat experimental data, as well as comparison to other published models. In addition, differences in performance are compared between the normal and categorically informed versions of the architecture

Neural Connectivity Underlying Reward and Emotion-Related Processing : Evidence From a Large-Scale Network Analysis

This work was supported by grants from the Economic and Social Research Council (ES/K013424/1) and the Leverhulme Trust (RPG-2019-010).Peer reviewedPublisher PD