The logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinitely imbalanced. In this paper, it is shown that this phenomenon is universal in a wide class of link functions on binomial regression. The proof relies on the extreme value theory. For the logit, p…
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48 results for “exponential link function”
Empirical study finds variance swap rate is affine in spot variance for S&P500 data.
problem Investigating the relationship between variance swap rate and spot variance.
method Empirical analysis using S&P500 data from 2006-2018, testing different models.
result Affine relationship between variance swap rate and spot variance is supported.
On Freedman's link packingsmath.GT
Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that these bounds seemed far too large. We show that the smallest of these "extravagantly…
Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approachmath.OC
We discuss an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the…
The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, , for any given crossing number .
result The number of alternating pretzel links grows exponentially with the crossing number.
We propose a Laplace approximation that creates a stochastic unit from any smooth monotonic activation function, using only Gaussian noise. This paper investigates the application of this stochastic approximation in training a family of Restricted Boltzmann Machines (RBM) that are closely linked to Bregman divergences.…
New MD algorithms using Tempesta logarithms for machine learning.
problem Optimization in machine learning with tailored hyperparameters.
method Developed Mirror Descent algorithms using Tempesta multi-parametric logarithms.
result Wide and flexible family of Mirror Descent and mirror-less updates.
Proposes a new framework for semi-supervised learning with theoretical support.
problem Lack of theoretical support for using predictions as pseudo-labels in deep SSL methods.
method D2 framework with repetitive reprediction (R2) strategy.
result R2-D2 method outperforms state-of-the-art methods by 5 percentage points on ImageNet.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
The paper introduces a new method for graph embedding using exponential family distributions.
problem Representing networks in a low dimensional latent space for various applications.
method Introduces the exponential family graph embedding model, generalizing random walk-based techniques to exponential family conditional distributions.
result The proposed techniques outperform existing methods in link prediction and node classification tasks.
We study the crossing number of links that are formed by edges of a triangulation T of the 3-sphere with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n^2. In general, this bound can not be replaced by a subexponential bound. However, if T is polytopal (resp. shellab…
A framework connects VAEs to GLMs for better model initialization and performance.
problem Understanding and optimizing loss function critical points in VAEs.
method Introducing a theoretical framework based on GLM and EDFs.
result Maximum likelihood initialization improves VAE performance.
Extends likelihood ratio exponential families to analyze various optimization methods.
problem Analyzing optimization methods like rate-distortion and information bottleneck.
method Linking geometric mixture paths to exponential families and using hypothesis testing.
result Provides a common mathematical framework for understanding these methods.
Study volume conjecture for links with multiple hyperbolic pieces.
problem Volume conjecture for links with more than one hyperbolic piece.
method Constructing infinite families of prime links, analyzing their complements, and using colored Jones polynomials and simplicial volume.
result Exponential growth rates of colored Jones polynomials capture the simplicial volume of link complements.
We investigate a class of hierarchical mixtures-of-experts (HME) models where exponential family regression models with generalized linear mean functions of the form psi(ga+fx^Tfgb) are mixed. Here psi(...) is the inverse link function. Suppose the true response y follows an exponential family regression model with mea…
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…
Study on entanglement complexity of confined ring polymers in lattice tubes.
problem Understanding the entanglement complexity of confined ring polymers in lattice tubes.
method Applied knot theory to extend and prove results about the complexity of 2SAPs.
result Proved that all but exponentially few size m 2SAPs have F complexity that grows at least linearly in m as m approaches infinity.
Let be a prime alternating link with crossings. We show that for each fixed , the number of genus incompressible surfaces in the complement of is bounded by a polynomial in . Previous bounds were exponential in .
Proposes an exponentially increasing step-size for faster parameter estimation in statistical models.
problem Slow convergence of gradient descent in locally convex loss functions.
method Exponentially increasing step-size in gradient descent algorithm.
result Converges linearly to optimal solution under homogeneous assumptions.
This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus general…
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…
In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second aut…
Polynomial bound on Reidemeister moves for each link type.
problem Recognizing whether a given link diagram represents a specific link type.
method Showed existence of a polynomial such that any two diagrams of a link type differ by at most Reidemeister moves.
result The problem of recognising a link type is in NP and can be completed in exponential time.
We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4-manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating pi_1-null surfaces in 4-manifolds. In our applications to link-homotopy, grope splitting serves a…
Let be a non-split prime alternating link with crossings. We show that for each fixed , the number of genus- Seifert surfaces for is bounded by an explicitly given polynomial in . The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponenti…
We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
We present an approximate Bayesian inference approach for estimating the intensity of an inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a latent marked Poisson process and Pólya--Gamma random variables w…
New model predicts links in community-based networks robustly.
problem Link prediction in community-based networks with local clustering errors.
method Markov Stochastic Block Model (MSBM) with Hidden Markov Model (HMM) predictions.
result Misclassification error decays exponentially with relevant signal-to-noise ratio (SNR).
Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of th…
A new method for estimating probabilities and risks using Markov processes.
problem Computational difficulties in classical importance sampling for latent Markov models.
method Proposes a new importance sampling framework that minimizes estimator variance.
result Shows logarithmic efficiency of the proposed estimator.
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
Jones polynomials for knots and links with many crossings calculated efficiently.
problem Computing Jones polynomials for knots and links with a large number of crossings.
method Calculating Tutte polynomials for associated graphs and evaluating with specific substitutions.
result Jones polynomials for knots and links with many crossings calculated efficiently.
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level ). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r…
Generic Unsupervised Optimization for a Latent Variable Model With Exponential Family Observablescs.LG
New LVMs optimize any exponential family distribution without specific assumptions.
problem Optimizing latent variable models with non-Gaussian observables.
method Generic optimization using EM approach for exponential family distributions.
result Concise parameter update equations applicable to various data types.
New method for efficient ERG fitting on large graphs.
problem Fitting non-trivial ERGs on large graphs.
method Fast matrix block-approximation techniques for dyadic independence.
result Models can generate networks with similar properties to observed networks.
We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant , in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit form…
We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. A linking curve is defined as a curve in the tangent space whose composition with the exponential map is tree formed. This key idea is used to define sutured manifolds. We prove first that any sutured manifold sati…
We present a novel method to reconstruct complex network from partial information. We assume to know the links only for a subset of the nodes and to know some non-topological quantity (fitness) characterising every node. The missing links are generated on the basis of the latter quan- tity according to a fitness model …
Nowozin \textit{et al} showed last year how to extend the GAN \textit{principle} to all -divergences. The approach is elegant but falls short of a full description of the supervised game, and says little about the key player, the generator: for example, what does the generator actually converge to if solving the GAN…
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the phys…
The paper calculates the asymptotics of quantum invariants for Whitehead chains.
problem Quantum invariants of Whitehead chains with colored clasps.
method Asymptotic analysis of colored Jones polynomials, considering limiting ratios of sequences.
result The exponential growth rate of invariants matches the hyperbolic volume of link complements.
Proves new concentration inequalities for sub-gaussian and sub-exponential variables.
problem Understanding functions of independent random variables better.
method Sub-gaussian and sub-exponential conditions, Rademacher complexities, Lipschitz function classes.
result Extension of Rademacher complexities to unbounded sub-exponential distributions.
Generative models unify heterogeneous data for multimodal fusion.
problem Learning effective representations of mixed numerical and categorical data.
method Bayesian approach with exponential family distributions and Laplace-Bernstein approximation.
result Generative models enable fusion of multimodal data for various machine learning tasks.
Parallel algorithm speeds up Jones polynomial computation.
problem Efficient computation of knot complexity measures.
method First parallel algorithm for exact Jones polynomial computation.
result Reduces computational time by an exponential factor.
We develop algorithms to learn non-linear dynamical systems without mixing assumptions.
problem Learning non-linear dynamical systems from dependent data.
method We introduce an offline algorithm and a one-pass streaming method with SGD-RER.
result Our methods achieve optimal or near-optimal performance for learning non-linear systems.