TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.
problem Modeling computation in a single step.
method Established Topological Kleene Field Theory (TKFT) as a new model of computation.
result Any computable function can be simulated in a single go of a dynamical system.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
Functions on trees are characterized by a system of PDEs.
problem Characterize functions that can be computed on tree structures.
method Developed a system of non-linear partial differential equations (PDEs) to describe functions on trees.
result Proved that these conditions are both necessary and sufficient for analytic and bit-valued functions.
The Cheap Gradient Principle (Griewank 2008) --- the computational cost of computing the gradient of a scalar-valued function is nearly the same (often within a factor of 5) as that of simply computing the function itself --- is of central importance in optimization; it allows us to quickly obtain (high dimensional) …
Survey on learning Boolean functions in computational theory.
problem Learning Boolean function classes in computational theory.
method Overview of known results in PAC and related models.
result Discussion of various learning results for Boolean functions.
New method calculates DMN log-likelihood faster.
problem Precise and fast computation of DMN log-likelihood.
method Derived a closed form expression using gamma function properties.
result Closed form calculation is faster with same accuracy.
A new method avoids partition function computation for Gibbs density estimation.
problem Estimating Gibbs density functions without partition function computation.
method Maximum Recovery MAP (MR-MAP) and least-action type potential.
result MR-MAP estimators solve optimization problem quickly using neural network.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
Efficiently computes quasiconcave envelope with limited data.
problem Approximating unknown quasiconcave function with partial information.
method Solves value problem and interpolation problem with polynomial and logarithmic LPs.
result Efficiently computes quasiconcave envelope with limited data.
Efficiently reduces computational burden of rollout acquisition functions in Bayesian optimization.
problem Expensive computation of rollout acquisition functions in Bayesian optimization.
method Combines quasi-Monte Carlo, common random numbers, and control variates to reduce computational burden. Formulates a policy-search approach to eliminate the need to optimize the rollout acquisition function.
result Significant reduction in computational burden of rollout acquisition functions.
Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
Machine learning impacts computational math, offering new functions approximations.
problem Machine learning's black box nature hinders further progress in computational math.
method Analyzes machine learning's impact on computational math and vice versa.
result Integrating computational math with machine learning can enhance both fields.
Paper uses deep learning to compute committor functions for rare events in complex systems.
problem Computing committor functions for low-temperature, high-dimensional systems is challenging.
method Combines deep learning, data sampling, and feature engineering.
result Achieves good performance on complex benchmark problems with rough energy landscapes.
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of sub…
Functional AD for Weil algebra computations.
problem Efficient computation of C∞-structures on Weil algebras. method Multivariate Tower Automatic Differentiation (AD) implementation.
result Functional AD for Weil algebra computation.
Researchers compute dimensions of GLN-skein modules for genus-one mapping tori.
problem Computing dimensions of GLN-skein modules for mapping tori.
method Explicit Euler product expansion of the skein partition function.
result Explicit computation of dimensions and generating function.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
OPAA estimates probability densities using functional analysis.
problem Estimating probability density functions efficiently and accurately.
method OPAA uses a parallelizable algorithm based on functional analysis to estimate probability distributions.
result OPAA provides an efficient method to estimate probability density functions and normalizing weights.
Framework for completing computational graphs using Gaussian Processes.
problem Completing computational graphs from incomplete data.
method Using Gaussian Processes to approximate unknown functions and recover unobserved variables.
result Efficiently completes computational graphs with fewer data points.
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
Modified BA algorithm computes RD and DR functions efficiently.
problem Computing rate-distortion and distortion-rate functions.
method A novel modification of the BA algorithm using Newton's method for root-finding.
result The modified algorithm converges to RD and DR function solutions with rate O(1/n) and provides ε-approximations. Zeta functions extended to nonorientable surfaces, order of vanishing computed.
problem Computing dynamical zeta functions for nonorientable surfaces.
method Simple argument extending microlocal proofs to nonorientable case.
result Order of vanishing of zeta function is the first Betti number.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
New method certifies neural network function space norms from point evaluations.
problem Certifying neural network function space norms from point evaluations alone.
method Combining interval arithmetic enclosures, adaptive marking/refinement, and quadrature-based aggregation.
result Certified computation of Lp, W1,p, and W2,p norms. Paper proves autodiff systems are correct for non-differentiable functions.
problem Correctness of autodiff systems for non-differentiable functions in deep learning.
method Investigation of PAP functions and introduction of intensional derivatives.
result Intensional derivatives always exist and coincide with standard derivatives for almost all inputs.
PDEs constrain smooth functions in neural networks.
problem Understanding functions computable by neural networks.
method Analyzing smooth hierarchical functions via PDEs.
result Established algebraic PDEs for smooth functions.
PHS optimizes hyperparameters in parallel for expensive computations.
problem Optimizing hyperparameters in computationally expensive tasks.
method Bayesian optimization for parallel hyperparameter search.
result Efficient hyperparameter optimization on multiple instances.
A new method for efficient computation of Knowledge Gradient in Bayesian optimization.
problem Efficient computation of the Knowledge Gradient for Bayesian optimization.
method One-shot Hybrid KG, a new approach combining previous ideas.
result The new method is cheap to compute and preserves theoretical properties of previous methods.
seMCD computes depth functions with statistical guarantees using sequential Monte Carlo.
problem Computing depth functions is computationally challenging, especially in high dimensions.
method Sequential Monte Carlo methodology with theoretical and empirical guarantees.
result The seMCD method provides accurate depth approximations with fewer samples than traditional methods.
The sophisticated structure of Convolutional Neural Network (CNN) allows for outstanding performance, but at the cost of intensive computation. As significant redundancies inevitably present in such a structure, many works have been proposed to prune the convolutional filters for computation cost reduction. Although ex…
Deep kernel learning for complex function modeling.
problem Modeling complex functions with line integral measurements.
method Gaussian process with neural networks for line integral data.
result Improved performance in computed tomography reconstruction.
In distributed function computation, each node has an initial value and the goal is to compute a function of these values in a distributed manner. In this paper, we propose a novel token-based approach to compute a wide class of target functions to which we refer as "Token-based function Computation with Memory" (TCM) …
New proof shows efficient ReLU networks for piecewise linear functions.
problem Existence of efficient ReLU neural networks for piecewise linear functions.
method Degree 1 triangulations of the relative homology class bounded by polyhedra.
result Existence of efficient ReLU neural networks for functions with compact support.
Human computation or crowdsourcing involves joint inference of the ground-truth-answers and the worker-abilities by optimizing an objective function, for instance, by maximizing the data likelihood based on an assumed underlying model. A variety of methods have been proposed in the literature to address this inference …
New formalism solves kinematical constraints in curved backgrounds and non-trivial states.
problem Solving kinematical constraints due to Weyl invariance in curved backgrounds and non-trivial states.
method Constructing Weyl covariant geometric objects and identifying them as building blocks of correlation functions.
result Exact agreement with thermal OPEs and holographic computations for thermal 2-point functions.
New method detects non-product domains using squeezing function.
problem Detecting non-product bounded pseudoconvex domains.
method New application of squeezing function and optimal estimates.
result Identifies new family of holomorphic homogeneous regular domains.
A framework for partially encrypted machine learning using functional encryption.
problem Performing machine learning on encrypted data without revealing sensitive information.
method Combining adversarial training and functional encryption to efficiently compute quadratic functions and prevent feature leakage.
result The proposed framework maintains high model accuracy while significantly improving data privacy.
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
Computes cusp cobordism groups for Morse functions on manifolds.
problem Understanding the cusp cobordism groups of Morse functions.
method Employed Levine's cusp elimination technique and created pairs of cusps along fold lines.
result Both unoriented and oriented cusp cobordism groups are cyclic of order two in even dimensions and infinite order in odd dimensions.
We analyze general model selection procedures using penalized empirical loss minimization under computational constraints. While classical model selection approaches do not consider computational aspects of performing model selection, we argue that any practical model selection procedure must not only trade off estimat…
We propose a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent applied to the problem of minimizing a strongly convex composite function represented as the sum of an average of a large number of smooth convex functions, and simple nonsmooth conve…
Establishes statistical and computational bounds for influence diagnostics.
problem Identifying influential datapoints or subsets in machine learning models.
method Finite-sample statistical bounds and computational complexity for influence functions and approximate maximum influence perturbations.
result Established statistical and computational guarantees for influence diagnostics.
We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…
Paper introduces a multi-stage influence function to track model predictions.
problem Improving natural language processing and computer vision performance.
method Develops a multi-stage influence function to track predictions from finetuned models back to pretraining data.
result Identifies pretraining examples contributing most to finetuning task predictions.
A new loss function for VAEs improves image quality and efficiency.
problem Training VAEs to generate realistic images requires a loss function that reflects human perception.
method Based on Watson's perceptual model, the loss function computes a weighted distance in frequency space, accounts for luminance and contrast masking, and is extended to color images.
result VAEs trained with the new loss function generated high-quality, less blurry images with fewer artifacts and less computational resources.
This technical report describes an efficient technique for computing the norm of the gradient of the loss function for a neural network with respect to its parameters. This gradient norm can be computed efficiently for every example.
Paper proposes new loss functions for training energy networks.
problem Challenges in computing gradients for training energy networks.
method Proposes generalized Fenchel-Young losses for efficient gradient computation.
result Demonstrates the calibration of excess risk for linear-concave energies.