We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof, details surrounding map covariant derivatives and a careful application of the DeTurck…
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48 results for “Tensor Tricks”
A popular trick for computing a pairwise co-occurrence matrix is the product of an incidence matrix and its transpose. We present an analog for higher order tuple co-occurrences using the face-splitting product, or alternately known as the transpose Khatri-Rao product. These higher order co-occurrences encode the commo…
Dualizable tensor categoriesmath.QA
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…
Study of skateboard flips as continuous curves in group.
problem Characterize skateboard flip tricks as continuous motions.
method Model flips as curves in , analyze lifts to , derive formulas.
result There are only four distinct flip tricks up to continuous deformation.
Nash's theorem proved with Günther's trick
problem Proving Nash's smooth embedding theorem
method Using Günther's trick
result Nash's theorem proved
Bayesian Tensor Network combines prior and data likelihood for efficient prediction and parameter estimation.
problem Overfitting and poor performance in Tensor Network models.
method Introduce prior distribution, use Laplace approximation for posterior predictive distribution, and propose stable initialization for parameter estimation.
result Reduces overfitting and improves performance of Tensor Network models.
Explains Conway's tangle trick and its mathematical origins.
problem Understanding the relationship between braids and elliptic curves.
method Discusses the tangle trick, its mathematical underpinnings, and historical context.
result Establishes the connection between braids and elliptic curves.
Kernelized cumulants improve statistical analysis in high-dimensional spaces.
problem Statistical analysis in high-dimensional spaces with low variance estimators.
method Extending cumulants to RKHS using tensor algebra and kernel trick.
result Kernelized cumulants provide new all-purpose statistics with computational tractability.
Unified framework for gradient estimation in combinatorial spaces.
problem Scaling relaxed gradient estimators to large combinatorial distributions.
method Introducing stochastic softmax tricks within the perturbation model framework.
result Stochastic softmax tricks improve model performance and discover more latent structure.
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
Geometric trick simplifies link homotopy and concordance.
problem Homotopy and concordance of links in homology spheres.
method Relative Whitney trick to remove double points.
result Links in homology spheres can be simplified to topologically slice links.
Outlier based Robust Principal Component Analysis (RPCA) requires centering of the non-outliers. We show a "bias trick" that automatically centers these non-outliers. Using this bias trick we obtain the first RPCA algorithm that is optimal with respect to centering.
A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learningcs.LG
The Gumbel-max trick and its extensions simplify sampling from categorical distributions in machine learning.
problem Sampling from categorical distributions with unnormalized probabilities.
method Extensions of the Gumbel-max trick for various applications.
result Simplified and efficient methods for sampling and gradient estimation.
A new gradient estimator for categorical distributions reduces bias and variance.
problem Intractability of gradients for categorical distributions in discrete latent variable models.
method CatLog-Derivative trick and IndeCateR gradient estimator.
result IndeCateR reduces bias and variance of gradients for categorical distributions.
Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.
problem Smooth triviality of Lie subalgebras and Lie ideals
method Establishing necessary and sufficient conditions and proving Moser's trick for foliations
result Direct proof of Moser's trick for foliations
Retail Product Image Classification is an important Computer Vision and Machine Learning problem for building real world systems like self-checkout stores and automated retail execution evaluation. In this work, we present various tricks to increase accuracy of Deep Learning models on different types of retail product …
We demonstrate that the uniqueness of solutions to a broad class of parabolic geometric evolution equations can be proven via a direct and essentially classical energy argument which avoids the DeTurck trick entirely. Previously, we have used a variation of this technique to give an alternative proof and slight extensi…
A triple-point Whitney trickmath.GT
Triple-point Whitney trick classifies ornaments of 3-manifolds.
problem Classifying ornaments of 3-manifolds in high dimensions.
method Triple-point Whitney trick applied to orientable manifolds.
result Classification of ornaments by the μ-invariant.
Alexander trick applied to homology spheres for manifold homeomorphisms.
problem Group of homeomorphisms of contractible manifolds.
method Strong uniqueness statement for one-sided h-cobordisms.
result Group of homeomorphisms is contractible for .
Embolic volume of compact manifolds is defined in terms of Berger's embolic inequality. In this paper, we show a result of relating embolic volume to the first Betti number. The proof relies on Gromov's covering argument appeared in systolic geometry. Berger called this method covering trick. We exploit and present mor…
Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
Bredon's trick helps extend local properties to global topological spaces.
problem Extending local properties to global topological spaces.
method Bredon's trick for local properties to global spaces.
result Bredon's trick allows for natural alternative demonstrations of classic results.
We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterizati…
We introduce a family of pairwise stochastic gradient estimators for gradients of expectations, which are related to the log-derivative trick, but involve pairwise interactions between samples. The simplest example of our new estimator, dubbed the fundamental trick estimator, is shown to arise from either a) introducin…
The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each time solving for the most likely configuration. We derive an entire family of r…
This work optimizes LLMs by reformulating attention networks and solving them in matrix multiplication time.
problem Optimizing the one-layer attention network objective function in LLMs.
method Provides an iterative greedy algorithm to train the loss function up to ε in time proportional to matrix multiplication.
result Solves the attention regression problem in matrix multiplication time.
An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of …
Derives a primal-dual MLSVD formulation for multilinear data.
problem Efficiently decompose multilinear data for signal analysis and deep learning.
method Kernelizable primal-dual formulation of MLSVD.
result Derives a new MLSVD formulation with computational advantages.
Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regre…
Study geometric flows of G2-structures, determining curvature and torsion invariants.
problem Investigate geometric flows of G2-structures and their invariants.
method Explicitly compute differential invariants, decompose curvature and torsion, analyze principal symbols.
result Established short-time existence and uniqueness for geometric flows of G2-structures.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
The reparameterization trick has become one of the most useful tools in the field of variational inference. However, the reparameterization trick is based on the standardization transformation which restricts the scope of application of this method to distributions that have tractable inverse cumulative distribution fu…
Boltzmann machines (BMs) are appealing candidates for powerful priors in variational autoencoders (VAEs), as they are capable of capturing nontrivial and multi-modal distributions over discrete variables. However, non-differentiability of the discrete units prohibits using the reparameterization trick, essential for lo…
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
Validation of the Replica Trick for Simple Modelscond-mat.dis-nn
We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic con…
Paper develops a weighted linearization approach for vector fields.
problem Linearizability of vector fields under weighted conditions.
method Formal Moser trick applied to power series, addressing weighted non-resonance condition.
result Formal Moser trick works over any field of characteristic zero.
Tricks adversarial attacks to target specific classes, improving classifier accuracy.
problem Recent adversarial defense approaches have failed to protect classifiers from untargeted attacks.
method Target Training defense tricks untargeted attacks into targeted attacks on designated classes, then derives the real class.
result 86.2% accuracy for CW-L2 (confidence=0) in CIFAR10, outperforming unsecured classifiers.
This paper solves a complex differential relation using a novel 'avoidance trick'.
problem Classifying tangent distributions satisfying non-involutivity conditions.
method Convex integration with an 'avoidance trick'.
result First example of a differential relation that is ample in some directions but not all.
A new network model combines features of DCBM, LSM, and β-model, using a cancellation trick for parameter estimation.
problem Challenging parameter fitting in network models.
method Introducing a cancellation trick to resolve parameter fitting issues in the logit-DCBM.
result R-SCORE significantly improves community detection over existing methods.
The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relative…
Low-variance gradient estimation is crucial for learning directed graphical models parameterized by neural networks, where the reparameterization trick is widely used for those with continuous variables. While this technique gives low-variance gradient estimates, it has not been directly applicable to discrete variable…
Proves existence of planar curves with specific curvature.
problem Existence of planar closed curves with prescribed curvature.
method Variational methods, adding a parameter, monotonicity trick.
result Existence of planar closed curves with prescribed curvature for some curvature functions.
The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.
problem Understanding quasi-alternating surgeries on asymmetric L-space knots.
method Using the Montesinos trick to confirm known surgeries.
result Confirmation of two quasi-alternating surgeries for each of 9 asymmetric L-space knots.
Improved diffusion bridge sampling with rKL-LD loss.
problem Improving sampling from unnormalized distributions using diffusion bridges.
method Employing the rKL-LD loss instead of the Log Variance (LV) loss for diffusion bridges.
result rKL-LD consistently outperforms LV loss in diffusion bridges.
We stabilize the Kumaraswamy distribution for efficient sampling and differentiation.
problem Numerical instabilities in the Kumaraswamy distribution's inverse CDF and log-pdf.
method Identified and resolved numerical issues, introduced a stabilized KS distribution.
result Stabilized Kumaraswamy distribution supports efficient sampling and differentiation.
A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertaintyphysics.flu-dyn
Proposes a new model for RANS simulations with uncertainty.
problem Uncertainty in Reynolds-averaged Navier-Stokes simulations.
method Data-driven closure model with aleatoric uncertainty, Bayesian formulation, sparse indirect data.
result Accurate probabilistic predictions, even in regions of model error.
The well-known Gumbel-Max trick for sampling from a categorical distribution can be extended to sample elements without replacement. We show how to implicitly apply this 'Gumbel-Top-' trick on a factorized distribution over sequences, allowing to draw exact samples without replacement using a Stochastic Beam Sea…
The paper proves symplectic neighbourhood theorems for stratified subspaces.
problem Finding symplectic neighbourhoods of stratified subspaces.
method Analogy with Weinstein's neighbourhood theorem, strong version of Moser's trick, and tubular neighbourhood theorem.
result Generalization of existing constructions for exotic Lagrangians.