A new method for deep Wishart processes improves kernel-based models.
problem Inference in deep Wishart processes is challenging due to the need for flexible distributions over positive semi-definite matrices.
method Developed a novel approach to flexible distributions over positive semi-definite matrices using the Bartlett decomposition of the Wishart probability density. Used this to create an approximate posterior for the DWP.
result Improved performance of inference in the DWP compared to DGP with equivalent prior.
Improved variational approximation for deep Wishart process models.
problem Improving predictive performance of deep Wishart process models.
method Generalizing the Bartlett decomposition of the Wishart distribution to allow linear combinations of rows and columns.
result Better predictive performance achieved with minimal additional computation cost.
We refine the analysis of hedging strategies for options under the SABR model carried out in [2]. In particular, we provide a theoretical justification of the empirical observation made in [2] that the modified delta ("Bartlett's delta") introduced there provides a more accurate and robust hedging strategy than the con…
This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator, in order to use composition techniques as Ninomiya and Victoir or Alfonsi. Doing so, we have found a remarkable splitting for Wishart proc…
A neural network improves DOA estimation from a single snapshot.
problem Estimating DOAs from a single snapshot with limited aperture.
method Deep learning architecture trained to generate high-resolution spatial spectrum.
result Our (SP)2-Net outperforms classical methods. We derive variance-optimal hedging strategies for SABR and rough Bergomi models.
problem Finding efficient hedging strategies in lognormal SABR and rough Bergomi models.
method Analytic expressions for variance-optimal hedging strategies and mean-square hedging errors.
result The variance-optimal hedging strategy in SABR coincides with Delta adjustment.
This is a brief technical note to clarify the state of lower bounds on regret for reinforcement learning. In particular, this paper: - Reproduces a lower bound on regret for reinforcement learning, similar to the result of Theorem 5 in the journal UCRL2 paper (Jaksch et al 2010). - Clarifies that the proposed proof of …
The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.
problem Understanding the structure and properties of symmetric matrices through Cholesky decompositions.
method Introducing cones of symmetric matrices, proving Cholesky-type factorizations, and showing geometric properties.
result Each symmetric matrix admits an uncountable family of Cholesky-type factorizations, and these cones are isometric Riemannian manifolds.
Generalizes string-net modular functors to non-spherical categories.
problem Extending string-net models to non-spherical categories.
method Using non-semisimple string-nets and Drinfeld centers.
result Equivalence between string-net and Lyubashenko modular functors.
We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the opti…
We present surrogate regret bounds for arbitrary surrogate losses in the context of binary classification with label-dependent costs. Such bounds relate a classifier's risk, assessed with respect to a surrogate loss, to its cost-sensitive classification risk. Two approaches to surrogate regret bounds are developed. The…
A generalized additive model (GAM, Hastie and Tibshirani (1987)) is a nonparametric model by the sum of univariate functions with respect to each explanatory variable, i.e., f(x)=∑fj(xj), where xj∈R is j-th component of a sample x∈Rp. In this paper, we introd…
In this paper we derive an efficient method for computing the indices associated with an asymptotically optimal upper confidence bound algorithm (MDP-UCB) of Burnetas and Katehakis (1997) that only requires solving a system of two non-linear equations with two unknowns, irrespective of the cardinality of the state spac…
Boosting is one of the most successful ideas in machine learning. The most well-accepted explanations for the low generalization error of boosting algorithms such as AdaBoost stem from margin theory. The study of margins in the context of boosting algorithms was initiated by Schapire, Freund, Bartlett and Lee (1998) an…
Uniform convergence of interpolators proven for Gaussian data.
problem Interpolation learning in high-dimensional linear regression with Gaussian data.
method Generic uniform convergence guarantee in terms of Gaussian width.
result Consistency of interpolators for minimum-norm and near-minimal-norm cases.
Identity testing for reversible Markov chains without symmetry assumption.
problem Identity testing of reversible Markov chains.
method Using distance notion from Daskalakis et al. [2018a], testing without symmetry assumption.
result It is possible to perform identity testing under weaker assumption of reversibility.
We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When the feature sequence is known at the beginning of the game, they provide…
We study the role of depth in training randomly initialized overparameterized neural networks. We give a general result showing that depth improves trainability of neural networks by improving the conditioning of certain kernel matrices of the input data. This result holds for arbitrary non-linear activation functions …
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as x↦WNWN−1⋯W1x) by minimizing the ℓ2 loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are…
Unified binary and multiclass margin-based classification methods.
problem No consensus on multiclass loss functions analogous to binary margin loss.
method Showed multiclass loss functions can be expressed in relative margin form.
result Extended classification-calibration result to multiclass.
We introduce SCAL, an algorithm designed to perform efficient exploration-exploitation in any unknown weakly-communicating Markov decision process (MDP) for which an upper bound c on the span of the optimal bias function is known. For an MDP with S states, A actions and Γ≤S possible next states, we prove a …
The scaled complex Wishart distribution is a widely used model for multilook full polarimetric SAR data whose adequacy has been attested in the literature. Classification, segmentation, and image analysis techniques which depend on this model have been devised, and many of them employ some type of dissimilarity measure…
Defines a new 3D TQFT from non-semisimple categories.
problem Developing a TQFT from non-semisimple categories.
method Generators and relations framework, decomposition of 3-manifolds.
result TQFT values on 3-manifolds match known invariants.
We propose to study the generalization error of a learned predictor h^ in terms of that of a surrogate (potentially randomized) predictor that is coupled to h^ and designed to trade empirical risk for control of generalization error. In the case where h^ interpolates the data, it is interesting to con…
New approach for online learning with adaptive adversaries, simpler and more effective.
problem Online learning with adaptive adversaries, especially in bandits and MDPs.
method Uses standard unbiased estimators and a simple increasing learning rate schedule, aided by logarithmically homogeneous self-concordant barriers and strengthened Freedman's inequality.
result First high-probability regret bounds for adversarial bandits and MDPs, resolving open problems.
This is a brief technical note to clarify some of the issues with applying the application of the algorithm posterior sampling for reinforcement learning (PSRL) in environments without fixed episodes. In particular, this paper aims to: - Review some of results which have been proven for finite horizon MDPs (Osband et a…
RL and DTSOC for final quadratic hedging performance studied.
problem Optimal hedging of European call options with and without transaction costs.
method Reinforcement Learning and Deep Trajectory-based Stochastic Optimal Control.
result RL and DTSOC perform similarly to variance-optimal hedging in various market models.
Researchers describe and compare decompositions of Poincaré duality pairs.
problem Understanding and comparing different decompositions of Poincaré duality pairs.
method Developed and described edge splittings of decompositions based on group properties.
result Compared decompositions with two other related decompositions.
Exciting new work on the generalization bounds for neural networks (NN) given by Neyshabur et al. , Bartlett et al. closely depend on two parameter-depenedent quantities: the Lipschitz constant upper-bound and the stable rank (a softer version of the rank operator). This leads to an interesting question of whether cont…
The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.
problem Decompositions of moduli spaces of vector bundles with fixed determinant of odd degree.
method Semiorthogonal decompositions, Grothendieck ring of varieties, mirror symmetry, graph potentials, Fukaya category.
result Evidence for a conjectural semiorthogonal decomposition of moduli spaces of rank 2 bundles with odd determinant.
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…
This paper generalizes octahedral decomposition to links in thickened surfaces.
problem Understanding the geometry of links in thickened surfaces.
method Octahedral decomposition of links in thickened surfaces.
result Nonpositive curvature of the complement and essential-ness of edges proved.
This paper studies recursive ensembles driven by Fibonacci updates, improving learning dynamics.
problem Improving learning dynamics in recursive ensemble learning.
method Develops second-order recursive architectures with Fibonacci-type update flows.
result Establishes global convergence conditions and generalization bounds for recursive ensembles.
Researchers compute Goeritz groups for all (1,1)-link decompositions.
problem Computing Goeritz groups for all (1,1)-link decompositions.
method Analyzing surface decompositions and isotopy classes of homeomorphisms.
result Computed Goeritz groups for all (1,1)-link decompositions.
Study concordance of decompositions from defining sequences in 3-sphere.
problem Understanding concordance and bordism of decompositions from defining sequences.
method Relate to invariants of toroidal decompositions and cobordism of homology manifolds.
result At least uncountably many concordance classes of decompositions in 3-sphere.
Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…
Study shows OAT decomposition generates unexplained profit and loss, while SU decompositions depend on risk factor order.
problem Understanding profit and loss attribution in financial markets.
method Used financial market data from 2003 to 2022 to compare OAT, SU, and ASU decompositions.
result SU decompositions are sensitive to risk factor order and cannot identify all relevant risk factors.
A new algorithm speeds up CP decomposition for large tensors.
problem Efficiently processing large-scale tensors in real-time.
method Randomized online CP decomposition (ROCP) algorithm.
result ROCP reduces computing time and memory usage significantly.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
Smooth 4-manifolds have simple horizontal decompositions.
problem Classifying smooth, closed, orientable 4-manifolds.
method Horizontal handlebody decomposition.
result Simplest horizontal decompositions classify closed 4-manifolds.
Let J1 be the real form of a complex simple Jordan algebra such that the automorphism group is F4(−20). By using some orbit types of F4(−20) on J1, for F4(−20), explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's Kε−Iwasawa decomp…
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, in a natural way its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…
New method uses random decompositions for high-dimensional Bayesian optimization.
problem Learning accurate decompositions for high-dimensional black-box functions.
method Data-independent random tree-based decomposition sampling.
result Random decomposition upper-confidence bound algorithm (RDUCB) yields significant empirical gains.
New varifold example shows decomposition failure.
problem Curvature varifolds cannot always be decomposed.
method Constructed a specific curvature varifold.
result Found a varifold with a non-preserved weak second fundamental form under decomposition.
Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
problem Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
method Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
result Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be k-manifolds (k=1,2), which generalize characterizations in the codimens…