Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
Researchers develop multi-utility representations for incomplete preferences linked to risk measures.
problem Handling incomplete preferences induced by set-valued risk measures.
method Established dual representations of set-valued risk measures to create parsimonious and well-behaved multi-utility representations.
result Unified dual representations of set-valued risk measures, linking them to scalar risk measures.
Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
Unified data representation learning improves non-parametric two-sample testing.
problem Improving non-parametric two-sample testing accuracy.
method Proposes RL-TST framework combining IRs and DRs for better test power.
result RL-TST outperforms existing methods by leveraging both IRs and DRs.
SRN improves set representations for relational reasoning.
problem Set permutational invariance limitations in existing approaches.
method Proposed a Set Refiner Network (SRN) to respect set invariance.
result Substantial gains in prediction performance and robustness on relational reasoning tasks.
Study genus-three Torelli maps and their fixed point sets in representation varieties.
problem Understanding fixed point sets and representation varieties of genus-three Torelli maps.
method Analyzing fixed point sets and representation varieties of powers of bounding pair maps.
result Determined the number of connected components of fixed point sets and representation varieties.
We study Anosov representations whose limit set has intermediate regularity, namely is a Lipschitz submanifold of a flag manifold. We introduce an explicit linear functional, the unstable Jacobian, whose orbit growth rate is integral on this class of representations. We prove that many interesting higher rank represent…
Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.
problem Understanding Anosov representations and their deformations.
method Cataclysm deformations based on twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
New representations for surface groups in PU(2,1) are stable and larger than convex cocompact ones.
problem Characterizing representations of surface groups in PU(2,1).
method Introducing simple-stable representations and proving their properties.
result The set of conjugacy classes of simple-stable representations is a domain of discontinuity strictly larger than convex cocompact representations.
Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.
problem Characterizing SL(2,R) representations on a once-punctured torus.
method Introduction of spectrum as a subset of projective measured laminations, analysis of dynamics of cocycles.
result Spectrum of a generic representation on a once-punctured torus is a Cantor set.
New method uses bi-level optimization to learn useful representations for imitation learning.
problem Learning useful representations for multiple tasks in imitation learning settings.
method Formulates representation learning as a bi-level optimization problem.
result Bi-level optimization framework provides sample complexity benefits for imitation learning.
Hidden symmetry of a G'-space X is defined by an extension of the G'-action on X to that of a group G containing G' as a subgroup. In this setting, we study the relationship between the three objects: (A) global analysis on X by using representations of G (hidden symmetry); (B) global analysis on X by using representat…
Survey on affine Anosov representations and their implications.
problem Generalizing Anosov representations to affine settings.
method Discussion and survey of existing work.
result Implications of affine Anosov representations.
This paper introduces a new feature learning technique based on error representation.
problem Learning high-level features for classification from diverse and imbalanced data.
method Inverse feature learning using error representation approach.
result Significantly better performance compared to state-of-the-art techniques.
We establish dual representations for systemic risk measures based on acceptance sets in a general setting. We deal with systemic risk measures of both "first allocate, then aggregate" and "first aggregate, then allocate" type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets …
Study on hyperconvex representations of surface groups and their geometric properties.
problem Understanding the geometry of hyperconvex representations of surface groups.
method Holomorphic extension of Ahlfors--Bers map and analysis of limit sets.
result Limit set has Hausdorff dimension 1 if and only if representation is in PSL(d,R).
The paper studies invariant convex sets in representations with nontrivial copolarity.
problem Understanding the face structure of invariant convex sets in representations with nontrivial copolarity.
method Proves that the face structure of an invariant convex set is determined by its intersection with a fat section, and that a face is exposed if and only if the corresponding face of the intersection is exposed.
result The face structure of invariant convex sets is completely determined by their intersections with fat sections, and exposed faces are preserved.
In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov repre…
New approach uses prior knowledge to improve neural network representations.
problem Difficulty in learning robust internal representations without a large training set.
method Incorporates prior information into a pre-trained reasoning module.
result Improves representation learning in diverse self-supervised learning settings.
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
Proposes MIGE for accurate MI gradient estimation in high-dimensional settings.
problem Intractability of MI in continuous and high-dimensional settings.
method Score estimation of implicit distributions for gradient estimation of MI.
result MIGE provides tight and smooth gradient estimation of MI in high-dimensional settings.
Study evaluates scalability and real-world impact of disentangled representations.
problem Scalability and real-world impact of disentangled representations.
method New high-resolution dataset and architectures for disentangled representation learning.
result Disentanglement predicts out-of-distribution task performance.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
New invariants defined for knots and links using quandle representations.
problem Defining new invariants for knots and links.
method Defined a family of quiver representations associated to finite quandles, abelian groups, and quandle 2-cocycles.
result Computed four new polynomial invariants for knots and links.
A framework compares image representations based on local geometry.
problem Comparing image representations based on global structure overlooks local differences.
method Quantify local geometry using Fisher information matrix and optimize differentiation with principal distortions.
result Identifies differences in local sensitivities between models.
Cataclysm deformations study Anosov representations and their convergence.
problem Understanding convergence of Anosov representations under deformation.
method Cataclysm deformation of Anosov representations using twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
Classifies Zariski closures of positive representations in Lie groups.
problem Classifying Zariski closures of positive representations in Lie groups.
method Classifies the Lie algebra of the Zariski closure of a discrete subgroup with specific properties.
result Obtains a new proof of Guichard's classification of Zariski closures of Hitchin representations.
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
Study on representations of four-punctured sphere group in hyperbolic spaces.
problem Understanding representations of the four-punctured sphere group.
method Investigation into simple-stable and Bowditch representations in Gromov-hyperbolic spaces.
result Simple-stable representations and Bowditch representations are equivalent.
New method for high-fidelity shape representations from raw data.
problem Creating accurate shape representations from raw data.
method A simple loss function encouraging neural network to vanish on input point cloud and have unit norm gradient.
result Our method produces high-fidelity, smooth, and natural zero level set surfaces.
This paper improves few-shot learning by reducing sample complexity using representation learning.
problem Reducing sample complexity for target tasks with limited data.
method Representation learning to pool all source task samples for target task learning.
result Representation learning can achieve substantial sample size reduction, bypassing the $Ω( rac{1}{T})$ barrier.
A test assesses the calibration of set-based epistemic uncertainty representations.
problem Evaluating the accuracy of set-based representations of epistemic uncertainty in machine learning.
method Proposes a novel statistical test to determine if a convex combination of predictions is calibrated, allowing instance-level variability.
result Demonstrates the benefits of capturing instance-level variability on synthetic and real-world experiments.
Proves equivalence of two types of representations of free groups in hyperbolic spaces.
problem Equivalence of two types of representations of free groups in hyperbolic spaces.
method Independent proof of equivalence for free groups of rank two in Gromov-hyperbolic spaces.
result Set of Bowditch representations equals set of primitive-stable representations.
Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not a…
State representation learning aims at learning compact representations from raw observations in robotics and control applications. Approaches used for this objective are auto-encoders, learning forward models, inverse dynamics or learning using generic priors on the state characteristics. However, the diversity in appl…
We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…
Contrastive learning harms minority group representations, affecting downstream tasks.
problem Representation harm in contrastive learning, especially affecting minority groups.
method Causal mediation analysis and stochastic block model explanation.
result Representation harm in contrastive learning is partly responsible for allocation harm in downstream tasks.
Convolutional layers in graph neural networks are a fundamental type of layer which output a representation or embedding of each graph vertex. The representation typically encodes information about the vertex in question and its neighbourhood. If one wishes to perform a graph centric task, such as graph classification,…
Contrastive unsupervised representation learning (CURL) is the state-of-the-art technique to learn representations (as a set of features) from unlabelled data. While CURL has collected several empirical successes recently, theoretical understanding of its performance was still missing. In a recent work, Arora et al. (2…
Study shows AD for neural nets with machine-representable numbers can be incorrect.
problem Correctness of AD for neural nets with machine-representable numbers.
method Analyzed two sets of parameters: incorrect and non-differentiable. Proved bounds and conditions for AD correctness.
result AD can be incorrect for machine-representable numbers, but provides a Clarke subderivative on non-differentiable set.
Robust reinforcement learning agents generalize well to out-of-distribution settings using pretrained representations.
problem Achieving sample-efficient reinforcement learning agents that generalize to real-world settings.
method Trained 240 representations and 10,000 RL policies on a simulated robotic setup, evaluating different pretrained VAE-based representations' effects on OOD generalization.
result Many reinforcement learning agents are surprisingly robust to realistic distribution shifts, including sim-to-real cases.
We compute the cohomology of a Fuchsian group of the second kind with coefficients in the hyperfunction vectors of the principal series representations of SL(2,R) supported on the limit set.
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that t…
In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.