Showing papers on "Class (philosophy) published in 2019"

Prediction and outlier detection in classification problems

Abstract: We consider the multi-class classification problem when the training data and the out-of-sample test data may have different distributions and propose a method called BCOPS (balanced and conformal optimized prediction sets). BCOPS constructs a prediction set $C(x)$ as a subset of class labels, possibly empty. It tries to optimize the out-of-sample performance, aiming to include the correct class as often as possible, but also detecting outliers $x$, for which the method returns no prediction (corresponding to $C(x)$ equal to the empty set). The proposed method combines supervised-learning algorithms with the method of conformal prediction to minimize a misclassification loss averaged over the out-of-sample distribution. The constructed prediction sets have a finite-sample coverage guarantee without distributional assumptions. We also propose a method to estimate the outlier detection rate of a given method. We prove asymptotic consistency and optimality of our proposals under suitable assumptions and illustrate our methods on real data examples.

Few-Shot Learning for Dermatological Disease Diagnosis.

Abstract: We consider the problem of clinical image classification for the purpose of aiding doctors in dermatological disease diagnosis. Diagnosis of dermatological conditions from images poses two major challenges for standard off-the-shelf techniques: First, the distribution of real-world dermatological datasets is typically long-tailed. Second, intra-class variability is large. To address the first issue, we formulate the problem as low-shot learning, where once deployed, a base classifier must rapidly generalize to diagnose novel conditions given very few labeled examples. To model intra-class variability effectively, we propose Prototypical Clustering Networks (PCN), an extension to Prototypical Networks (Snell et al., 2017) that learns a mixture of “prototypes” for each class. Prototypes are initialized for each class via clustering and refined via an online update scheme. Classification is performed by measuring similarity to a weighted combination of prototypes within a class, where the weights are the inferred cluster responsibilities. We demonstrate the strengths of our approach in effective diagnosis on a realistic dataset of dermatological conditions.

From Braces to Hecke algebras & Quantum Groups

Abstract: We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices.

Motivic Voice-Leading in Selected Piano Cycles of Johannes Brahms

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Topological fields with a generic derivation

Abstract: We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_\delta$-extension $T_{\delta}^*$ by a generic derivation $\delta$. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has $\mathcal{L}$-open core (i.e., every $\mathcal{L}_\delta$-definable open set is $\mathcal{L}$-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of $T$ such as relative elimination of field sort quantifiers, NIP and distality also transfer to $T_\delta^*$. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of $p$-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

Intensional Protocols for Dynamic Epistemic Logic

Abstract: In dynamical multi-agent systems, agents are controlled by protocols. In choosing a class of formal protocols, an implicit choice is made concerning the types of agents, actions and dynamics representable. This paper investigates one such choice: An intensional protocol class for agent control in dynamic epistemic logic (DEL), called ‘DEL dynamical systems’. After illustrating how such protocols may be used in formalizing and analyzing information dynamics, the types of epistemic temporal models that they may generate are characterized. This facilitates a formal comparison with the only other formal protocol framework in dynamic epistemic logic, namely the extensional ‘DEL protocols’. The paper concludes with a conceptual comparison, highlighting modeling tasks where DEL dynamical systems are natural.

Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems.

Abstract: We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization by several examples, in particular a nonlinear Schrodinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.

Stochastic Volterra integral equations and a class of first order stochastic partial differential equations

Abstract: We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued \Levy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

Groups with ALOGTIME-hard word problems and PSPACE-complete compressed word problems.

Abstract: We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's groups) we prove that the compressed word problem (which is equivalent to the circuit evaluation problem) is $\mathsf{PSPACE}$-complete.

A Robust Class of Linear Recurrence Sequences

Abstract: We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.

Separation and covering for group based concatenation hierarchies

Abstract: Concatenation hierarchies are natural classifications of regular languages. All such hierarchies are built through the same construction process: one starts from an initial, specific class of languages (the basis) and builds new levels using two generic operations. Concatenation hierarchies have gathered a lot of interest since the early 70s, notably thanks to an alternate logical definition: each concatenation hierarchy can be defined as the quantification alternation hierarchy within a variant of first-order logic over words (while the hierarchies differ by their bases, the variants differ by their set of available predicates). Our goal is to understand these hierarchies. A typical approach is to look at two decision problems: membership and separation. In the paper we are interested in the latter, which is more general. For a class of languages $C$ , C-separation takes two regular languages as input and asks whether there exists a third one in $C$ including the first one and disjoint from the second one. Settling whether separation is decidable for the levels within a given concatenation hierarchy is among the most fundamental and challenging questions in formal language theory. In all prominent cases, it is open, or answered positively for low levels only. Recently, a breakthrough was made using a generic approach for a specific kind of hierarchies: those with a finite basis. In this case. separation is always decidable for levels 1/2. 1 and 3/2. Our main theorem is similar but independent: we consider hierarchies with possibly infinite bases, but that contain only group languages. An example is the group hierarchy introduced by Pin and Margolis: its basis consists of all group languages. Another example is the quantifier alternation hierarchy of first-order logic with modular predicates FO( $ , MOD): its basis consists of the languages that count the length of words modulo some number. Using a generic approach, we show that for any such hierarchy, if separation is decidable for the basis, then it is decidable as well for levels 1/2, 1 and 3/2 (we actually solve a more general problem called covering). This complements the aforementioned result nicely: all bases considered in the literature are either finite or made of group languages. Thus, one may handle the lower levels of any prominent hierarchy in a generic way.

Determinantal probability measures on Grassmannians

Abstract: We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting.

Model-free Bootstrap for a General Class of Stationary Time Series

Abstract: A model-free bootstrap procedure for a general class of stationary time series is introduced. The theoretical framework is established, showing asymptotic validity of bootstrap confidence intervals for many statistics of interest. In addition, asymptotic validity of one-step ahead bootstrap prediction intervals is also demonstrated. Finite-sample experiments are conducted to empirically confirm the performance of the new method, and to compare with popular methods such as the block bootstrap and the autoregressive (AR)-sieve bootstrap.

Cellular categories and stable independence

Abstract: We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.

A class of recursive optimal stopping problems with applications to stock trading

Abstract: In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.

Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans.

Abstract: We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.

The Intrinsic Normal Cone For Artin Stacks

Abstract: We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.

The zero mass problem for Klein-Gordon equations

Abstract: We are interested in the global solutions to a class of Klein-Gordon equations, and particularly in the unified time decay results with respect to the possibly vanishing mass parameter. We give for the first time a rigorous proof, which relies on both the flat foliation and the hyperboloidal foliation of the Minkowski spacetime. In order to take advantages of both foliations, an iteration procedure is used.

Towards Deterministic Decomposable Circuits for Safe Queries

Abstract: There exist two approaches for exact probabilistic inference of UCQs on tuple-independent databases. In the extensional approach, query evaluation is performed within a DBMS by exploiting the structure of the query. In the intensional approach, one first builds a representation of the lineage of the query on the database, then computes the probability of the lineage. In this paper we propose a new technique to construct lineage representations as deterministic decomposable circuits in PTIME. The technique can apply to a class of UCQs that has been conjectured to separate the complexity of the two approaches. We test our technique experimentally, and show that it succeeds on all the queries of this class up to a certain size parameter, i.e., over $20$ million queries.

Fault Tolerant Network Constructors.

Abstract: In this work, we consider adversarial crash faults of nodes in the network constructors model $[$Michail and Spirakis, 2016$]$. We first show that, without further assumptions, the class of graph languages that can be (stably) constructed under crash faults is non-empty but small. In particular, if an unbounded number of crash faults may occur, we prove that (i) the only constructible graph language is that of spanning cliques and (ii) a strong impossibility result holds even if the size of the graphs that the protocol outputs in populations of size $n$ need only grow with $n$ (the remaining nodes being waste). When there is a finite upper bound $f$ on the number of faults, we show that it is impossible to construct any non-hereditary graph language. On the positive side, by relaxing our requirements we prove that: (i) permitting linear waste enables to construct on $n/(2f)-f$ nodes, any graph language that is constructible in the fault-free case, (ii) partial constructibility (i.e. not having to generate all graphs in the language) allows the construction of a large class of graph languages. We then extend the original model with a minimal form of fault notifications. Our main result here is a fault-tolerant universal constructor: We develop a fault-tolerant protocol for spanning line and use it to simulate a linear-space Turing Machine $M$. This allows a fault-tolerant construction of any graph accepted by $M$ in linear space, with waste $min\{n/2+f(n),\; n\}$, where $f(n)$ is the number of faults in the execution. We then prove that increasing the permissible waste to $min\{2n/3+f(n),\; n\}$ allows the construction of graphs accepted by an $O(n^2)$-space Turing Machine, which is asymptotically the maximum simulation space that we can hope for in this model. Finally, we show that logarithmic local memories can be exploited for a no-waste fault-tolerant simulation of any such protocol.

The Neumann problem for a class of fully nonlinear elliptic partial differential equations

Abstract: In this paper, we establish a global $C^2$ estimates to the Neumann problem for a class of fullly nonlinear elliptic equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problems.

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

Abstract: We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

Solving ill-structured problems mediated by online- discussion forums: Mass customisation of learning

Abstract: This paper was published in the 2019 proceedings of the 36TH International Conference on Innovation, Practice and Research in the Use of Educational Technologies in Tertiary Education, ASCILITE 2019: Personalised Learning. Diverse Goals. One Heart, held at the Singapore University of Social Sciences, 2 - 5 December 2019.

Solvable Hypergroups and a Generalization of Hall's Theorems on Finite Solvable Groups to Association Schemes

Abstract: We generalize Philip Hall's celebrated theorems on finite solvable groups to scheme theory. Our result is based on a series of results on hypergroups.

Mutation-finite quivers with real weights

Abstract: We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by reflections, and thus gives rise to a notion of a $Y$-seed. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.

Towards Deterministic Decomposable Circuits for Safe Queries

Abstract: There exist two approaches for exact probabilistic inference of UCQs on tuple-independent databases. In the extensional approach, query evaluation is performed within a DBMS by exploiting the structure of the query. In the intensional approach, one first builds a representation of the lineage of the query on the database, then computes the probability of the lineage. In this paper we propose a new technique to construct lineage representations as deterministic decomposable circuits in PTIME. The technique can apply to a class of UCQs that has been conjectured to separate the complexity of the two approaches. We test our technique experimentally, and show that it succeeds on all the queries of this class up to a certain size parameter, i.e., over $20$ million queries.

Representations of surface groups with universally finite mapping class group orbit

Abstract: Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2-2g-n<0$. Let $$\rho: \pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f: \Sigma_{g', n'}\to \Sigma_{g, n}$, the orbit of (the isomorphism class of) $f^*(\rho)$ under the mapping class group $MCG(\Sigma_{g',n'})$ of $\Sigma_{g',n'}$ is finite. Then we show that $\rho$ has finite image. The result is motivated by the Grothendieck-Katz $p$-curvature conjecture, and gives a reformulation of the $p$-curvature conjecture in terms of isomonodromy.

A Large Scale Approach to Decomposition Spaces

Abstract: Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces, shearlet spaces and modulation spaces are well-known decomposition spaces. In this paper we focus on the geometric aspects of decomposition spaces and utilize that these are naturally captured by the large scale properties of a metric space, the covered space, associated to a covering of a set. We demonstrate that decomposition spaces constructed out of quasi-isometric covered spaces have many geometric features in common. The notion of geometric embedding is introduced to formalize the way one decomposition space can be embedded into another decomposition space while respecting the geometric features of the coverings. Some consequences of the large scale approach to decomposition spaces are (i) comparison of coverings of different sets, (ii) study of embeddings of decomposition spaces based on the geometric features and the symmetries of the coverings and (iii) the use of notions from large scale geometry, such as asymptotic dimension or hyperbolicity, to study the properties of decomposition spaces.