We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential in…
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48 results for “joint invariants”
Study joint invariants on symplectic spaces, extending group and space variations.
problem Computing joint invariants on linear symplectic spaces.
method Review and extend previous work on group and space variations, relate to differential invariants.
result New computations and extensions of joint invariants.
We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss geometric properties of joint invariants of Killing two-tensors defined in the …
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
The paper tackles joint learning of linear systems, improving accuracy with pooled data.
problem Estimating transition matrices of multiple related linear systems more accurately.
method Developed novel techniques to bound estimation errors and establish high probability bounds for singular values.
result Significant gains in accuracy achieved by pooling data across systems.
A constructive version of the Frobenius integrability theorem -- that can be programmed effectively -- is given. This is used in computing invariants of groups of low ranks and recover examples from a recent paper of Boyko, Patera and Popoyvich \cite{BPP}.
Study presents a method to induce a generalized neural network from joint group invariant functions.
problem Encoding rule of neural network internal data representation.
method Systematic method using joint group invariant function on data-parameter domain.
result Induces a generalized neural network and its inverse operator (ridgelet transform).
Pairwise counter-monotonicityq-fin.RM
Study on pairwise counter-monotonicity, a type of negative dependence.
problem Understanding and quantifying extremal negative dependence structures.
method Established stochastic representation and invariance property; showed implications and connections.
result Pairwise counter-monotonicity implies negative association and joint mix dependence.
The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a connected sum is the smash product (in the sense of equivariant spectra) of the invaria…
We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using v…
BCOV invariant and blow-upmath.DG
Extends BCOV invariant to pairs of Calabi-Yau manifolds and pluricanonical divisors.
problem Extend BCOV invariant to new geometric pairs.
method Extend BCOV invariant to pairs , study blow-up behavior.
result Results imply birational Calabi-Yau manifolds have the same BCOV invariant.
First I will explain my motivation to introduce the -invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of -invariants to several areas in mathematics. Finally, I will present two optimal inequalities …
The paper proposes a method to create domain-invariant representations using Wasserstein distance.
problem Domain shifts in training data affect machine learning model performance across different domains.
method The method combines classification/regression losses with a GAN-type discriminator to minimize the Wasserstein distance between domains.
result The approach produces the highest minimum classification accuracy and most invariant representation across domains.
Absolute Whitehead torsionmath.AT
We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\bA$ from an element of $\widetilde{K}^{iso}_1(\bA)$ to an element of the absolute group $K_1^{iso}(\bA)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invari…
Proposes a new algorithm to estimate invariant subspaces across multilayer networks.
problem Estimating invariant subspaces across heterogeneous multiple networks.
method Bias-corrected joint spectral embedding algorithm that recursively calibrates diagonal bias and iteratively updates the subspace estimator.
result Established entrywise subspace perturbation bound and entrywise eigenvector central limit theorem for the algorithm.
We review a construction of hyperkahler metrics proposed in joint work of Davide Gaiotto, Greg Moore and the author. A key ingredient in this construction is a collection of integer "DT invariants" obeying the wall-crossing formula of Kontsevich-Soibelman.
On the Learnability of Knot Invariants: Representation, Predictability, and Neural Similaritymath.GT
The study analyzes neural network predictions of knot invariants and finds that braid representations work best.
problem Understanding and predicting knot invariants using neural networks.
method Investigated different knot representations and invariants, proposed a cosine similarity score.
result Braid representations are best for predicting knot invariants, and some invariants are easier to learn than others.
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kähler class, is used to define and characterize a special class of totally geodesic embeddings, called "tight embeddings". In addition, special is…
We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant is of infinite…
Due to the ability of deep neural nets to learn rich representations, recent advances in unsupervised domain adaptation have focused on learning domain-invariant features that achieve a small error on the source domain. The hope is that the learnt representation, together with the hypothesis learnt from the source doma…
Rozansky-Witten theorymath.QA
Rozansky and Witten proposed in 1996 a family of new three-dimensional topological quantum field theories, indexed by compact (or asymptotically flat) hyperkaehler manifolds. As a byproduct they proved that hyperkaehler manifolds also give rise to Vassiliev weight systems. These may be thought of as invariants of hyper…
BOtied optimizes multiple objectives using copulas and CDF indicators.
problem Joint optimization of multiple competing objectives.
method Proposes BOtied, a new acquisition function based on the CDF.
result BOtied outperforms state-of-the-art MOBO acquisition functions.
New method disentangles style features from data augmentations.
problem Difficulty in deducing which data attributes are 'style' and should be discarded.
method Structured data augmentation with multiple style embedding spaces, maximizing joint entropy.
result Empirically demonstrates benefits on synthetic and real-world data.
Validates conformal prediction for network data under non-uniform sampling.
problem Validity of conformal prediction for network data under non-representative sampling.
method Interprets sampling mechanisms as selection rules, studies validity conditional on selection events, uses permutation invariance and joint exchangeability.
result Finite-sample validity of conformal prediction for certain selection events and asymptotic validity for random walk sampling.
Optimizes VAE hyperparameters for efficient training and manifold discovery.
problem Efficiently optimizing hyperparameters in VAEs for complex data.
method Latent Bayesian Optimization (zBO) for hyperparameter trajectory optimization.
result Demonstrated improved performance in finding joint rotationally invariant representations.
Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. They include such well-known and important models as the Kepler potential, Calogero-Moser model, and harmonic oscillator, as well as its inte…
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector …
In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formu…
This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relatio…
Unsupervised Domain Adaptation aims to learn a model on a source domain with labeled data in order to perform well on unlabeled data of a target domain. Current approaches focus on learning \textit{Domain Invariant Representations}. It relies on the assumption that such representations are well-suited for learning the …
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
Paper develops new conformal prediction methods for sum or average of unknown labels.
problem Uncertainty quantification in joint distributions of random variables.
method Introduces novel conformal prediction methods for sum or average of unknown labels.
result Validates the proposed method for sum or average of unknown labels under permutation invariant assumptions.
This is a note for the conference proceedings Topological and Geometrical Problems related to Quantum Field Theory. We summarize our joint work with Dai about eta invariants on manifolds with boundary. Then we apply these results to prove the curvature and holonomy formulas for the natural connection on the determinant…
Domain adaptation framework identifies latent variables for target distribution identifiability.
problem Unsupervised domain adaptation without identifiable joint distribution of features and labels.
method Formulated latent variable model with invariant and changing components, constrained domain shift to influence only changing components.
result Joint distribution of data and labels in target domain is identifiable under mild conditions.
Proposes novel losses for fine-grained categorical domain adaptation.
problem Fine-grained alignment of categories across domains in unsupervised domain adaptation.
method Joint category-domain classifier with adversarial training losses for both domain and category levels, and vicinal domain adaptation.
result Achieves state-of-the-art performance on benchmark datasets.
In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type . The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants of …
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extend…
In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of fla…
Unveiling Invariant and Transferable Latent Factors Across Heterogeneous Environments via ATLASmath.ST
ATLAS separates invariant and transferable latent factors across diverse environments.
problem Transfer learning and robust prediction in heterogeneous environments.
method ATLAS leverages invariance principle to disentangle latent factors and uses auxiliary labels for robust prediction.
result Near-oracle performance and robust transferable prediction in new environments.
The paper explores the relationship between joint mixability and negative dependence structures.
problem Understanding the connection between joint mixability and various negative dependence concepts.
method Analyzes the properties of joint mixes and their relation to negative dependence structures.
result Derives necessary and sufficient conditions for a joint mix to be negatively dependent.
We show that we can release the rigidity of the skew Howe duality process for knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine case, corresponding to looking at tan…
FJS method improves multinomial classification accuracy.
problem Improving multinomial classification accuracy under dataset shift.
method Derive FJS representation and propose alternative methods.
result Factorizable joint shift is not fully identifiable without additional assumptions.
On evaluation of joint risk for non-negative multivariate risks under dependence uncertaintyq-fin.RM
Paper proposes a new method to evaluate joint risk under uncertainty.
problem Evaluating joint risk of multiple insurance risks under dependence uncertainty.
method Axiomatic approach to scalar and vector-valued distortion joint risk measures.
result Established a new scalar distortion joint risk measure with positive homogeneity.
Introduces joint Shapley values to measure feature importance in models.
problem Measuring the importance of feature sets in machine learning models.
method Extends Shapley's axioms to measure a set of features' average contribution to a model's prediction.
result Joint Shapley values provide unique insights and are more consistent with local intuitions.
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…
These are notes from lectures given at the Clay Institute Summer School on "Floer homology, gauge theory and low-dimensional topology" (Budapest, 2004). The first part describes as background some of the geometry of symplectic fibre bundles and their monodromy. The second part, overviewing joint work with Paul Seidel, …
Study proposes a new model for joint survival annuity valuation.
problem Valuation of joint survival annuities and options.
method Linear-rational Wishart mortality model based on stochastic matrix affine process.
result Derives closed-form expression for joint survival annuity and option.
Proposes a method to predict responses from covariates over time.
problem Predicting responses from covariates with changing conditional distributions over time.
method Invariant Subspace Decomposition (ISD) framework that splits the conditional distribution into time-invariant and time-dependent components.
result The decomposition can be used for zero-shot and time-adaptation prediction tasks.