The subject of graph convexity is well explored in the literature, the so-called interval convexities above all. In this work, we explore the cycle convexity, whose interval function is $I(S) = S \cup \{u \mid G[S \cup \{u\}]$ has a cycle containing $u\}$. In this convexity, we prove that the decision problems associated to the parameters rank and convexity number are in \NP-complete and \W[1]-hard when parameterized by the solution size. We also prove that to determine whether the percolation time of a graph is at least $k$ is \NP-complete, but polynomial for cacti or when $k\leq2$
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Through an uncertainty quantification (UQ) perspective, we show that score-based generative models (SGMs) are provably robust to the multiple sources of error in practical implementation. Our primary tool is the Wasserstein uncertainty propagation (WUP) theorem, a model-form UQ bound that describes how the $L^2$ error from learning the score function propagates to a Wasserstein-1 ($\mathbf{d}_1$) ball around the true data distribution under the evolution of the Fokker-Planck equation. We show how errors due to (a) finite sample approximation, (b) early stopping, (c) score-matching objective choice, (d) score function parametrization expressiveness, and (e) reference distribution choice, impact the quality of the generative model in terms of a $\mathbf{d}_1$ bound of computable quantities. The WUP theorem relies on Bernstein estimates for Hamilton-Jacobi-Bellman partial differential equations (PDE) and the regularizing properties of diffusion processes. Specifically, PDE regularity theory shows that stochasticity is the key mechanism ensuring SGM algorithms are provably robust. The WUP theorem applies to integral probability metrics beyond $\mathbf{d}_1$, such as the total variation distance and the maximum mean discrepancy. Sample complexity and generalization bounds in $\mathbf{d}_1$ follow directly from the WUP theorem. Our approach requires minimal assumptions, is agnostic to the manifold hypothesis and avoids absolute continuity assumptions for the target distribution. Additionally, our results clarify the trade-offs among multiple error sources in SGMs.
In this paper, we present a logic for conditional strong historical necessity in branching time and apply it to analyze a nontheological version of Lavenham's argument for future determinism. Strong historical necessity is motivated from a linguistical perspective, and an example of it is ``If I had not gotten away, I must have been dead''. The approach of the logic is as follows. The agent accepts ontic rules concerning how the world evolves over time. She takes some rules as indefeasible, which determine acceptable timelines. When evaluating a sentence with conditional strong historical necessity, we introduce its antecedent as an indefeasible ontic rule and then check whether its consequent holds for all acceptable timelines. The argument is not sound by the logic.
The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given $\ell_2$ error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.
Digital credentials represent a cornerstone of digital identity on the Internet. To achieve privacy, certain functionalities in credentials should be implemented. One is selective disclosure, which allows users to disclose only the claims or attributes they want. This paper presents a novel approach to selective disclosure that combines Merkle hash trees and Boneh-Lynn-Shacham (BLS) signatures. Combining these approaches, we achieve selective disclosure of claims in a single credential and creation of a verifiable presentation containing selectively disclosed claims from multiple credentials signed by different parties. Besides selective disclosure, we enable issuing credentials signed by multiple issuers using this approach.
Latent variable models serve as powerful tools to infer underlying dynamics from observed neural activity. However, due to the absence of ground truth data, prediction benchmarks are often employed as proxies. In this study, we reveal the limitations of the widely-used 'co-smoothing' prediction framework and propose an improved few-shot prediction approach that encourages more accurate latent dynamics. Utilizing a student-teacher setup with Hidden Markov Models, we demonstrate that the high co-smoothing model space can encompass models with arbitrary extraneous dynamics within their latent representations. To address this, we introduce a secondary metric -- a few-shot version of co-smoothing. This involves performing regression from the latent variables to held-out channels in the data using fewer trials. Our results indicate that among models with near-optimal co-smoothing, those with extraneous dynamics underperform in the few-shot co-smoothing compared to 'minimal' models devoid of such dynamics. We also provide analytical insights into the origin of this phenomenon. We further validate our findings on real neural data using two state-of-the-art methods: LFADS and STNDT. In the absence of ground truth, we suggest a proxy measure to quantify extraneous dynamics. By cross-decoding the latent variables of all model pairs with high co-smoothing, we identify models with minimal extraneous dynamics. We find a correlation between few-shot co-smoothing performance and this new measure. In summary, we present a novel prediction metric designed to yield latent variables that more accurately reflect the ground truth, offering a significant improvement for latent dynamics inference.
We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph $G$ and integers $d$ and $k$ decides in time $f(k,d)\cdot n^c$ for a computable function~$f$ and constant $c$ whether the elimination distance of $G$ to the class of degree $d$ graphs is at most $k$.
We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming variables. First, we give a sufficient and necessary condition for A to guarantee the following generalisation of Hilbert's Basis Theorem: every polynomial ideal which is equivariant, i.e. invariant under renaming of variables, is finitely generated. Second, we develop an extension of classical Buchberger's algorithm to compute a Gr\"obner basis of a given equivariant ideal. This implies decidability of the membership problem for equivariant ideals. Finally, we sketch upon various applications of these results to register automata, Petri nets with data, orbit-finitely generated vector spaces, and orbit-finite systems of linear equations.
In this work, we compute the lower bound of the integrality gap of the Metric Steiner Tree Problem (MSTP) on a graph for some small values of number of nodes and terminals. After debating about some limitations of the most used formulation for the Steiner Tree Problem, namely the Bidirected Cut Formulation, we introduce a novel formulation, that we named Complete Metric formulation, tailored for the metric case. We prove some interesting properties of this formulation and characterize some types of vertices. Finally, we define a linear program (LP) by adapting a method already used in the context of the Travelling Salesman Problem. This LP takes as input a vertex of the polytope of the CM relaxation and provides an MSTP instance such that (a) the optimal solution is precisely that vertex and (b) among all of the instances having that vertex as its optimal solution, the selected instance is the one having the highest integrality gap. We propose two heuristics for generating vertices to provide inputs for our procedure. In conclusion, we raise several conjectures and open questions.
For a set of robots (or agents) moving in a graph, two properties are highly desirable: confidentiality (i.e., a message between two agents must not pass through any intermediate agent) and efficiency (i.e., messages are delivered through shortest paths). These properties can be obtained if the \textsc{Geodesic Mutual Visibility} (GMV, for short) problem is solved: oblivious robots move along the edges of the graph, without collisions, to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means there is a shortest path (i.e., a ``geodesic'') between each pair of robots along which no other robots reside. In this work, we optimally solve GMV on finite hexagonal grids $G_k$. This, in turn, requires first solving a graph combinatorial problem, i.e. determining the maximum number of mutually visible vertices in $G_k$.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
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