Quantum Sp(4) representation shows asymptotic faithfulness.
problem Proving asymptotic faithfulness for a new family of quantum representations.
method Generalized from skein SU(2)_k representations proof.
result First example of asymptotic faithfulness outside A_n family.
Quantum SU(n) representations are asymptotically faithful with norm estimates.
problem Asymptotic faithfulness of quantum SU(n) representations of mapping class groups.
method Peak sections in Kodaira embedding and parallell transport of projective connection.
result Norm estimates and asymptotic faithfulness of quantum SU(n) representations.
We generalize the asymptotic faithfulness of the skein quantum SU(2) representations of mapping class groups of orientable closed surfaces to skein SU(3). Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…
We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…
For each oriented surface Σ of genus g we study a limit of quantum representations of the mapping class group arising in TQFT derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on $\boH(Σ)$, the space of regular f…
Improved causal discovery methods for large graphs without strict assumptions.
problem Sub-optimal solutions due to faithfulness assumption violations.
method Super-structure estimation and local search strategies.
result The proposed method scales to hundreds of nodes with high accuracy.
Study on faithfulness of Burau representation for Artin-Tits groups.
problem Determining faithfulness of Burau representation for Artin-Tits groups.
method Algorithmic approach based on categorical methods.
result Burau representation is not faithful in specific cases.
Proposes a weaker faithfulness assumption for causal discovery.
problem Violation of the faithfulness assumption in causal discovery.
method Proposes a new assumption called 2-adjacency faithfulness and a modified Grow and Shrink algorithm.
result Proves the correctness of the modified algorithm under weaker assumptions.
Proves Burau representation is faithful for 4-strand braids.
problem Proving the Burau representation is faithful for 4-strand braids.
method Used ideas from Moody, Long, Long-Paton, and Bigelow.
result Burau representation is faithful for n = 4.
Bayesian networks are typically faithful, with implications for causal inference.
problem Determining the typicality of faithfulness in Bayesian networks.
method Analysis of Bayesian networks over a given DAG, parametrized by conditional exponential families, and nonparametric conditional densities.
result The faithful Bayesian networks are dense and open with respect to the total variation metric, extending existing results for specific classes of Bayesian networks.
For each fixed n>=2 we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus g>=2 is determined by the sequence of quantum SU(n) representations, when one considers all levels. That this is the case is a consequence of our asymptotic faithfulness property. We here provide explici…
Two new metrics assess LLM faithfulness and entropy, improving model reliability.
problem Evaluating the accuracy of LLMs in generating coherent responses.
method Proposes SF and SEP metrics based on information theory and thermodynamics.
result High SF and SEP scores indicate more faithful LLM responses.
In this paper we provide a review of asymptotic results of Toeplitz operators and their applications in TQFT. To do this we review the differential geometric construction of the Hitchin connection on a prequantizable compact symplectic manifold. We use asymptotic results relating the Hitchin connec- tion and Toeplitz o…
Extends Jones' construction to Thompson's group F and link homology.
problem Link homology and Thompson's group F.
method Asymptotic construction of (n,n)-tangles in Thompson's group F. result Asymptotically faithful association of (n,n)-tangles with elements of Thompson's group F. Study evaluates feature ranking methods' faithfulness in ML models, improving with dimensionality reduction.
problem Quantifying and improving the faithfulness of feature ranking methods in ML models.
method Evaluation of multiple feature ranking methods, including SHAP, LIME, ALE variance, and LR coefficients, using permutation importance as a baseline.
result Dimensionality reduction improves the faithfulness of feature ranking methods, making permutation importance the most faithful method.
The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$,…
The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton imp…
Paper relaxes faithfulness assumption for causal discovery using interventions.
problem Violation of faithfulness assumption in natural systems leads to incorrect causal structure identification.
method Use intervention-immediacy faithfulness assumption to identify causal structures with hard interventions.
result Interventions contain information about causal structure that can identify causal structures when faithfulness is violated.
In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group G. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of G. We prove a result which allows us to establish the same conclusion when G is assumed …
Geodesic patterns, shears, and Anosov representations of the modular group.
problem Understanding representations of the modular group into Isom(X).
method Analyzing geodesic patterns, shears, and foliations.
result The Barbot component is homeomorphic to R^2 x [0,∞), with interior and boundary properties.
We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a question of W. Goldman. We also prove the existence of faithful representations into PU…
The Burau representation of 3-strand braid group modulo p is determined and shown to be faithful for small p.
problem Determining the faithfulness of the Burau representation of B3 modulo p. method Algorithm and proof for faithfulness, solving Salter's question for all p.
result The Burau representation of B3 modulo p is faithful for p≤13 and for all p. New algorithm estimates causal effects for non-Gaussian data.
problem Estimating causal effects in non-Gaussian distributions.
method Generalized k-Triangle Faithfulness Assumption and Edge Estimation Algorithm.
result Uniformly consistent estimates of causal effects.
Study improves causal model discovery by relaxing assumptions for latent variables.
problem Discovering causal models with latent variables under weaker assumptions.
method Uses Answer Set Programming to discover semi-Markovian causal models with weakened Faithfulness assumption.
result Weakened Faithfulness assumption preserves power and speeds up discovery for causal models with latent variables.
Skein algebra action is faithful if quantum parameter isn't a root of 1.
problem Faithfulness of geometric action of skein algebras.
method Examined the Kauffman bracket skein algebra and its action on handlebody modules.
result The action is faithful if and only if the quantum parameter is not a root of 1.
The study evaluates how well local explanations align with model predictions.
problem Capturing the faithfulness of local explanations to model predictions.
method Introducing consistency and sufficiency as properties, and developing quantitative measures and estimators.
result Quantitative measures of consistency and sufficiency depend on test-time data distribution.
This paper deals with chain graphs under the alternative Andersson-Madigan-Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. We also show that the extension of Meek's conjecture to AMP chain graphs does n…
We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston …
The paper proposes new interpretability paradigms to improve model faithfulness.
problem Improving the accuracy of explanations for complex models.
method Examining and evolving existing paradigms, proposing new models.
result Three new paradigms for interpretability are presented.
This paper shows how to learn causal structures without faithfulness assumption.
problem Learning causal structures from non-experimental data without faithfulness assumption.
method Define modes of convergence and prove that achieving the best mode requires following the standard design practice.
result A learning algorithm that achieves the best mode of convergence must follow the standard design practice.
Measures faithfulness of LLM explanations to reveal hidden biases and misleading claims.
problem LLM explanations can misrepresent the model's reasoning process, leading to over-trust and misuse.
method Defines faithfulness in terms of concept influence and uses counterfactuals and Bayesian models to estimate it.
result Can quantify and discover interpretable patterns of unfaithfulness in LLM explanations.
Counterexamples show Salter's question on Burau image is negative for n=4.
problem Conditions for a matrix to be in the Burau image of B4. method Analyzing the central quotient and using counterexamples.
result The central quotient of the Burau image group does not coincide with the central quotient of a specific subgroup of the unitary group for n=4. We analyze two braid group representations and their reductions modulo p.
problem Analyzing the faithfulness of Gassner and Burau representations modulo p.
method Topological methods, including GL_n(Z[t_1^±1, ..., t_n^±1]) and GL_n(Z_p) reductions.
result The Gassner representation is faithful modulo p for all n and p > 1.
New framework improves interpretability of neural networks by enforcing explicitness, faithfulness, and stability.
problem Lack of interpretability in complex machine learning models.
method Proposes self-explaining neural networks with explicitness, faithfulness, and stability as desiderata. Designs models in stages, progressively generalizing linear classifiers to complex models, and enforces these properties via regularization.
result Experimental results show that the proposed framework offers a promising direction for reconciling model complexity and interpretability.
Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.
We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.
This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the regular Gaussian distributions that factorize with respect to a chain graph G with d parameters have positive Lebesgue measure with respect to Rd, whereas those that factorize with respect…
ID-ExpO fine-tunes neural networks for more faithful explanations.
problem Improving the faithfulness of explanations for complex machine learning models.
method Differentiable insertion/deletion metric-aware regularizers for optimization.
result Fine-tuned predictors produce more faithful explanations.
Investigates spectral problems in Finsler geometry with novel dimension pairs.
problem Spectral problems in Finsler geometry due to nonlinearity of the Finsler-Laplacian operator.
method Introduced 'faithful dimension pairs' to define the spectrum of compact reversible Finsler metrics.
result Provided upper and lower bounds for eigenvalues, extending previous results.
Finite index subgroups of certain groups cannot act faithfully on the circle.
problem Finite index subgroups of specific groups cannot act faithfully on the circle.
method Analyzing C1 actions on the circle for finite index subgroups of mapping class groups, automorphism groups, and outer automorphism groups. result No orientation preserving C1 action of finite index subgroups of these groups on the circle can be faithful. We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.
The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.
Paper relaxes faithfulness assumption for MB discovery in k-order Markov blanket.
problem Learning graphical Markov blanket from data under faithfulness assumption violations.
method k-order relaxation of faithfulness assumption; k-OMB algorithm.
result k-OMB recovers MB under true and empirical faithfulness violations.
The paper quantizes concatenated noisy vectors to a common cluster center, improving performance over naive methods.
problem Clustering concatenated noisy vectors from multiple sources.
method Asymptotic analysis of weighted sum of distances to a common cluster center.
result The clustering approach outperforms naive methods in terms of average distortion.
Improved lower bounds for faithful linear representations of mapping class groups.
problem Finding the minimum dimension of faithful linear representations of mapping class groups.
method Using finer study of commutation relations and specific pants decompositions to show representations must kill certain subgroups.
result Established lower bounds of 4g−3 for faithful representations of mapping class groups of genus g≥7. Method produces faithful representations of Garside groups and torus knot groups.
problem Representations of Garside groups and torus knot groups.
method Method to produce faithful representations in GL2(C[t±1, q±1])
result Explicit construction of representations for G(n,m) and torus knot groups.
We prove that the hyperelliptic mapping class group of a nonorientable surface of genus g≥4 has a faithful linear representation of dimension g2−1 over R.
In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful repres…