IA Scholar Query: Borel Equivalence Relations and Classifications of Countable Models.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 28 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Diagnosing and Fixing Manifold Overfitting in Deep Generative Models
https://scholar.archive.org/work/j72pv6vph5hclotfvbqricoxgy
Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a low-dimensional manifold embedded in high-dimensional ambient space. In this paper we investigate the pathologies of maximum-likelihood training in the presence of this dimensionality mismatch. We formally prove that degenerate optima are achieved wherein the manifold itself is learned but not the distribution on it, a phenomenon we call manifold overfitting. We propose a class of two-step procedures consisting of a dimensionality reduction step followed by maximum-likelihood density estimation, and prove that they recover the data-generating distribution in the nonparametric regime, thus avoiding manifold overfitting. We also show that these procedures enable density estimation on the manifolds learned by implicit models, such as generative adversarial networks, hence addressing a major shortcoming of these models. Several recently proposed methods are instances of our two-step procedures; we thus unify, extend, and theoretically justify a large class of models.Gabriel Loaiza-Ganem, Brendan Leigh Ross, Jesse C. Cresswell, Anthony L. Cateriniwork_j72pv6vph5hclotfvbqricoxgyMon, 28 Nov 2022 00:00:00 GMTGamma-convergence of a nonlocal perimeter arising in adversarial machine learning
https://scholar.archive.org/work/6j4v32hxzrauhljuizknlilvxu
In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowski type to a local anisotropic perimeter. The nonlocal model describes the regularizing effect of adversarial training in binary classifications. The energy essentially depends on the interaction between two distributions modelling likelihoods for the associated classes. We overcome typical strict regularity assumptions for the distributions by only assuming that they have bounded BV densities. In the natural topology coming from compactness, we prove Gamma-convergence to a weighted perimeter with weight determined by an anisotropic function of the two densities. Despite being local, this sharp interface limit reflects classification stability with respect to adversarial perturbations. We further apply our results to deduce Gamma-convergence of the associated total variations, to study the asymptotics of adversarial training, and to prove Gamma-convergence of graph discretizations for the nonlocal perimeter.Leon Bungert, Kerrek Stinsonwork_6j4v32hxzrauhljuizknlilvxuMon, 28 Nov 2022 00:00:00 GMTValid and efficient imprecise-probabilistic inference with partial priors, I. First results
https://scholar.archive.org/work/uhcxf2a6pfd4pb6rozcugukmae
Between Bayesian and frequentist inference, it's commonly believed that the former is for cases where one has a prior and the latter is for cases where one has no prior. But the prior/no-prior classification isn't exhaustive, and most real-world applications fit somewhere in between these two extremes. That neither of the two dominant schools of thought are suited for these applications creates confusion and slows progress. A key observation here is that "no prior information" actually means no prior distribution can be ruled out, so the classically-frequentist context is best characterized as every prior. From this perspective, it's now clear that there's an entire spectrum of contexts depending on what, if any, partial prior information is available, with Bayesian (one prior) and frequentist (every prior) on opposite extremes. This paper ties the two frameworks together by formally treating those cases where only partial prior information is available using the theory of imprecise probability. The end result is a unified framework of (imprecise-probabilistic) statistical inference with a new validity condition that implies both frequentist-style error rate control for derived procedures and Bayesian-style coherence properties, relative to the given partial prior information. This new theory contains both the Bayesian and frequentist frameworks as special cases, since they're both valid in this new sense relative to their respective partial priors. Different constructions of these valid inferential models are considered, and compared based on their efficiency.Ryan Martinwork_uhcxf2a6pfd4pb6rozcugukmaeSat, 26 Nov 2022 00:00:00 GMTThe Local Gross-Prasad Conjecture over Archimedean Local Fields
https://scholar.archive.org/work/fab5h75shzbt3j2del2djllrx4
Following the approach of C. Moeglin and J.L.-Waldspurger, this article completes the proof for the generic cases of the local Gross-Prasad conjecture over Archimedean local fields.Cheng Chenwork_fab5h75shzbt3j2del2djllrx4Wed, 23 Nov 2022 00:00:00 GMTConcentration of the missing mass in metric spaces
https://scholar.archive.org/work/rdax7oar3rajvoeybdaivb2gma
We study the estimation and concentration on its expectation of the probability to observe data further than a specified distance from a given iid sample in a metric space. The problem extends the classical problem of estimation of the missing mass in discrete spaces. We give some estimators for the conditional missing mass and show that estimation of the expected missing mass is difficult in general. Conditions on the distribution, under which the Good-Turing estimator and the conditional missing mass concentrate on their expectations are identified. Applications to anomaly detection, coding, the Wasserstein distance between true and empirical measure and simple learning bounds are sketched.Andreas Maurerwork_rdax7oar3rajvoeybdaivb2gmaTue, 22 Nov 2022 00:00:00 GMTDescent modulus and applications
https://scholar.archive.org/work/hcdbtlfasrblxfr4a6ksnl3kle
The norm of the gradient ∇f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, ∂f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f |(x). In this work we propose an axiomatic definition of descent modulus T [f ](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.Aris Daniilidis, Laurent Miclo, David Salaswork_hcdbtlfasrblxfr4a6ksnl3kleMon, 21 Nov 2022 00:00:00 GMTMean-field limit of non-exchangeable systems
https://scholar.archive.org/work/spdwnfdkpredpknwjg2mlih2sq
This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.Pierre-Emmanuel Jabin, David Poyato, Juan Solerwork_spdwnfdkpredpknwjg2mlih2sqFri, 18 Nov 2022 00:00:00 GMTOn compact extensions of tracial W^*-dynamical systems
https://scholar.archive.org/work/tt55yacpwne2jdsv5tc647nqse
We establish several classification results for compact extensions of tracial W^*-dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg-Zimmer type dichotomies on the L^2-level. Although in general an adequate generalization of the Furstenberg-Zimmer tower of intermediate compact extensions doesn't seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host-Kra-Ziegler tower for any ergodic action of a countable abelian group.Asgar Jamneshan, Pieter Spaaswork_tt55yacpwne2jdsv5tc647nqseThu, 17 Nov 2022 00:00:00 GMTCointegration with Occasionally Binding Constraints
https://scholar.archive.org/work/pxsamvgai5adtdmjge42bc7uoi
In the literature on nonlinear cointegration, a long-standing open problem relates to how a (nonlinear) vector autoregression, which provides a unified description of the short- and long-run dynamics of a collection of time series, can generate 'nonlinear cointegration' in the profound sense of those series sharing common nonlinear stochastic trends. We consider this problem in the setting of the censored and kinked structural VAR (CKSVAR), which provides a flexible yet tractable framework within which to model time series that are subject to threshold-type nonlinearities, such as those arising due to occasionally binding constraints, of which the zero lower bound (ZLB) on short-term nominal interest rates provides a leading example. We provide a complete characterisation of how common linear and nonlinear stochastic trends may be generated in this model, via unit roots and appropriate generalisations of the usual rank conditions, providing the first extension to date of the Granger-Johansen representation theorem from a linear to a nonlinear setting, and thereby giving the first successful treatment of the open problem. The limiting common trend processes include regulated, censored and kinked Brownian motions, none of which have previously appeared in the literature on cointegrated VARs. Our results and running examples illustrate that the CKSVAR is capable of supporting a far richer variety of long-run behaviour than is a linear VAR, in ways that may be particularly useful for the identification of structural parameters. En route to establishing our main results, we also develop a set of sufficient conditions for the processes generated by a CKSVAR to be stationary, ergodic, and weakly dependent.James A. Duffy, Sophocles Mavroeidis, Sam Wycherleywork_pxsamvgai5adtdmjge42bc7uoiThu, 17 Nov 2022 00:00:00 GMTInfinite volume Gibbs states and metastates of the random field mean-field spherical model
https://scholar.archive.org/work/fwwdkwhf4rcrlm74g5dsmdbd3e
For the discrete random field Curie-Weiss models, the infinite volume Gibbs states and metastates have been investigated and determined for specific instances of random external fields. In general, there are not many examples in the literature of non-trivial limiting metastates for discrete or continuous spin systems. We analyze the infinite volume Gibbs states of the mean-field spherical model, a model of continuous spins, in a general random external field with independent identically distributed components with finite moments of some order larger than four and non-vanishing variances of the second moments. Depending on the parameters of the model, we show that there exist three distinct phases: ordered ferromagnetic, ordered paramagnetic, and spin glass. In the ordered ferromagnetic and ordered paramagnetic phases, we show that there exists a unique infinite volume Gibbs state almost surely. In the spin glass phase, we show the existence of chaotic size dependence, provide a construction of the Aizenman-Wehr metastate, and consider both the convergence in distribution and almost sure convergence of the Newman-Stein metastates. The limiting metastates are non-trivial and their structure is universal due to the presence of Gaussian fluctuations and the spherical constraint.Kalle Koskinenwork_fwwdkwhf4rcrlm74g5dsmdbd3eWed, 16 Nov 2022 00:00:00 GMTOn the power of euclidean division: Lower bounds for algebraic machines, semantically
https://scholar.archive.org/work/hkgtdopqlfa2noluiatrajuft4
This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that 𝚖𝚊𝚡𝚏𝚕𝚘𝚠, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations". We further improve on this result by showing that euclidean division cannot be computable in polylogarithmic time using division on the reals, pinpointing that euclidean division provides a significant boost in expressive power. On top of showing this new separation result, we show our method captures previous lower bounds results from the literature: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to Ptime in the real case, and Mulmuley's lower bounds for prams without bit operations.Thomas Seiller and Luc Pellissier and Ulysse Léchinework_hkgtdopqlfa2noluiatrajuft4Tue, 15 Nov 2022 00:00:00 GMTNon-Sequential Decentralized Stochastic Control Revisited: Static Reducibility and Causality
https://scholar.archive.org/work/3kijv6xf5ng6tekyd4zw4vnvjm
In decentralized stochastic control (or stochastic team theory) and game theory, if there is a pre-defined order in a system in which the agents act, the system is called sequential, otherwise it is non-sequential. Much of the literature on stochastic control theory, such as studies on the existence analysis, approximation methods, and on dynamic programming or other analytical or learning theoretic methods, have focused on sequential systems. The static reduction method for sequential control problems (via change of measures or other techniques), in particular, has been shown to be very effective in arriving at existence, structural, approximation and learning theoretic results. Many practical systems, however, are non-sequential where the order of agents acting is random, and dependent on the realization of solution paths and prior actions taken. The study of such systems is particularly challenging as tools applicable for sequential models are not directly applicable. In this paper, we will study static reducibility of non-sequential stochastic control systems, including by change of measure methods. We revisit the notion of Causality (a definition due to Witsenhausen and which has been refined by Andersland and Tekenetzis), and provide an alternative representation using imaginary agents. Via this representation, we show that Causality, under an absolute continuity condition, allows for an equivalent static model whose reduction is policy-independent. This facilitates much of the stochastic analysis available for sequential systems to also be applicable for non-sequential systems. We further show that under more relaxed conditions on the model, such as solvability, such a reduction, when possible at all, is policy-dependent and thus has limited utility. We will also present a further reduction method for partially nested causal non-sequential systems.Ryan Simpson, Serdar Yukselwork_3kijv6xf5ng6tekyd4zw4vnvjmTue, 15 Nov 2022 00:00:00 GMTCo-spectral radius, equivalence relations and the growth of unimodular random rooted trees
https://scholar.archive.org/work/uqusqoutk5cmnpynlu2a5yqodu
We define the co-spectral radius of inclusions 𝒮≤ℛ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on G/H for inclusion H≤ G of groups. The almost sure existence of the sampling exponent is already new for i.i.d. percolation clusters on countable groups. For the proof, we develop a general method called 2-3-method that is based on the mass-transport principle. As a byproduct, we show that the growth of a unimodular random rooted tree of bounded degree always exists, assuming its upper growth passes a critical threshold. This complements Timar's work who showed the possible nonexistence of growth below this threshold. We also show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for Property (T) and hyperfinite relations.Miklós Abert, Mikolaj Fraczyk, Ben Hayeswork_uqusqoutk5cmnpynlu2a5yqoduMon, 14 Nov 2022 00:00:00 GMTUnified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
https://scholar.archive.org/work/eijaw33nkjcy5iqnvf5hjiznmq
Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces.Davide Barilari, Andrea Mondino, Luca Rizziwork_eijaw33nkjcy5iqnvf5hjiznmqMon, 14 Nov 2022 00:00:00 GMTCompactness criteria for Stieltjes function spaces and applications
https://scholar.archive.org/work/462q3zbdl5fj3pszsxetrncuwm
In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli-Arzel\'a and Kolmogorov-Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.Francisco J. Fernández and F. Adrián F. Tojo and Carlos Villanuevawork_462q3zbdl5fj3pszsxetrncuwmMon, 14 Nov 2022 00:00:00 GMTCan you take Akemann–Weaver's _ℵ_1 away?
https://scholar.archive.org/work/pbonjblqqbau7ebeeysa5tvsje
By Glimm's dichotomy, a separable, simple C^*-algebra has continuum-many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all C^*-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem, using Jensen's diamond axiom _ℵ_1, a powerful diagonalization principle that implies the Continuum Hypothesis (𝖢𝖧). By a result of Rosenberg, a separably represented simple C^*-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple C^*-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's _ℵ_1, denoted ^𝖢𝗈𝗁𝖾𝗇, that holds in the original Cohen's model for the negation of 𝖢𝖧. We also prove that ^𝖢𝗈𝗁𝖾𝗇 suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given C^*-algebra with a prescribed action on its space of pure states.Daniel Calderón, Ilijas Farahwork_pbonjblqqbau7ebeeysa5tvsjeSun, 13 Nov 2022 00:00:00 GMTThe (twisted) Eberlein convolution of measures
https://scholar.archive.org/work/lb3ayhpzpfg4pidja5qvpf2lmq
In this paper, we study the properties of the Eberlein convolution of measures and introduce a twisted version of it. For functions we show that the twisted Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (twisted) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (twisted) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (twisted) Eberlein convolution.Daniel Lenz, Timo Spindeler, Nicolae Strungaruwork_lb3ayhpzpfg4pidja5qvpf2lmqSun, 13 Nov 2022 00:00:00 GMTEmpirical Risk Minimization with Generalized Relative Entropy Regularization
https://scholar.archive.org/work/jt7d56ngkbbq5ek7cnpbhulm2q
The empirical risk minimization (ERM) problem with relative entropy regularization (ERM-RER) is investigated under the assumption that the reference measure is a σ-finite measure instead of a probability measure. This assumption leads to a generalization of the ERM-RER (g-ERM-RER) problem that allows for a larger degree of flexibility in the incorporation of prior knowledge over the set of models. The solution of the g-ERM-RER problem is shown to be a unique probability measure mutually absolutely continuous with the reference measure and to exhibit a probably-approximately-correct (PAC) guarantee for the ERM problem. For a given dataset, the empirical risk is shown to be a sub-Gaussian random variable when the models are sampled from the solution to the g-ERM-RER problem. Finally, the sensitivity of the expected empirical risk to deviations from the solution of the g-ERM-RER problem is studied. In particular, the expectation of the absolute value of sensitivity is shown to be upper bounded, up to a constant factor, by the square root of the lautum information between the models and the datasets.Samir M. Perlaza, Gaetan Bisson, Iñaki Esnaola, Alain Jean-Marie, Stefano Riniwork_jt7d56ngkbbq5ek7cnpbhulm2qSat, 12 Nov 2022 00:00:00 GMTContinuity of characteristics of composite quantum systems
https://scholar.archive.org/work/kztfh6rkt5bmnmux53rhrkeqvm
General methods of quantitative and qualitative continuity analysis of characteristics of composite quantum systems are described. Several modifications of the Alicki-Fannes-Winter method are considered, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. A new approximation method for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This method allows us to prove several general results (Simon-type dominated convergence theorem, the theorem about preserving continuity under convex mixtures, etc.). Uniform continuity bounds and local continuity conditions for basic characteristics of composite quantum systems are presented. Along with the results obtained earlier by different authors, a number of new results proved by the proposed methods are described.M.E.Shirokovwork_kztfh6rkt5bmnmux53rhrkeqvmFri, 11 Nov 2022 00:00:00 GMTHilbert Curve Projection Distance for Distribution Comparison
https://scholar.archive.org/work/iorwlsty3ngtphibvrvuj45hpi
Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the empirical HCP distance with the L_p cost in the d-dimensional space converges to its population counterpart at a rate of no more than O(n^-1/2max{d,p}). To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.Tao Li, Cheng Meng, Hongteng Xu, Jun Yuwork_iorwlsty3ngtphibvrvuj45hpiFri, 11 Nov 2022 00:00:00 GMT