Weaker Notion Research Articles

Egalitarian roommate allocations: Complexity and stability

We study two roommate assignment problems, called Ordinal Roommate Allocation and Cardinal Roommate Allocation, where students have preferences over roommates, rooms have varying capacities, and the goal is to maximize the minimum payoff of the students (under two distinct notions of payoff). Both problems are NP-hard when room sizes are unrestricted. In contrast, the Ordinal Roommate Allocation problem becomes tractable when the maximum room capacity is fixed, while the Cardinal Roommate Allocation problem remains NP-hard even with bounded room capacity and number of preferences. We then analyze the problems through the lens of stability, considering envy-freeness and a weaker notion we call swap-resistance. Not all instances guarantee an envy-free outcome, and it is shown to be NP-hard to determine which ones do. However, swap-resistance is always achievable using an efficient algorithm. We discuss connections and distinctions between our work and existing research about utilitarian matchings and stable roommate problems.

Going Deeper, Generalizing Better: An Information-Theoretic View for Deep Learning.

Deep learning has transformed computer vision, natural language processing, and speech recognition. However, two critical questions remain obscure: 1) why do deep neural networks (DNNs) generalize better than shallow networks and 2) does it always hold that a deeper network leads to better performance? In this article, we first show that the expected generalization error of neural networks (NNs) can be upper bounded by the mutual information between the learned features in the last hidden layer and the parameters of the output layer. This bound further implies that as the number of layers increases in the network, the expected generalization error will decrease under mild conditions. Layers with strict information loss, such as the convolutional or pooling layers, reduce the generalization error for the whole network; this answers the first question. However, algorithms with zero expected generalization error do not imply a small test error. This is because the expected training error is large when the information for fitting the data is lost as the number of layers increases. This suggests that the claim "the deeper the better" is conditioned on a small training error. Finally, we show that deep learning satisfies a weak notion of stability and provides some generalization error bounds for noisy stochastic gradient decent (SGD) and binary classification in DNNs.

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via ℓp,q regularization ( 0⩽q⩽1⩽p ) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the ℓp,q regularization, and apply it to establish an oracle property and a recovery bound for the ℓp,q regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the ℓp,q regularization problems, provide analytical formulas for proximal subproblems of certain specific ℓp,q regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Communication-Efficient Multi-Party Computation for RMS Programs

Despite much progress, general-purpose secure multi-party computation (MPC) with active security may still be prohibitively expensive in settings with large input datasets. This particularly applies to the secure evaluation of graph algorithms, where each party holds a subset of a large graph. Recently, Araki et al. (ACM CCS '21) showed that dedicated solutions may provide significantly better efficiency if the input graph is sparse. In particular, they provide an efficient protocol for the secure evaluation of “message passing” algorithms, such as the PageRank algorithm. Their protocol's computation and communication complexity are both O ~ ( M · B ) instead of the O ( M 2 ) complexity achieved by general-purpose MPC protocols, where M denotes the number of nodes and B the (average) number of incoming edges per node. On the downside, their approach achieves only a relatively weak security notion; 1 -out-of- 3 malicious security with selective abort. In this work, we show that PageRank can instead be captured efficiently as a restricted multiplication straight-line (RMS) program, and present a new actively secure MPC protocol tailored to handle RMS programs. In particular, we show that the local knowledge of the participants can be leveraged towards the first maliciously-secure protocol with communication complexity linear in M , independently of the sparsity of the graph. We present two variants of our protocol. In our communication-optimized protocol, going from semi-honest to malicious security only introduces a small communication overhead, but results in quadratic computation complexity O ( M 2 ) . In our balanced protocol, we still achieve a linear communication complexity O ( M ) , although with worse constants, but a significantly better computational complexity scaling with O ( M · B ) . Additionally, our protocols achieve security with identifiable abort and can tolerate up to n − 1 corruptions.

The geometry of C1,α flat isometric immersions

We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$ . In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha >2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

On the efficiency and fairness of deferred acceptance with single tie-breaking

As a random allocation rule for indivisible object allocation under weak priorities, deferred acceptance with single tie-breaking (DA-STB) is not ex-post constrained efficient. We first observe that it also fails to satisfy equal-top fairness, which requires that two agents be assigned their common top choice with equal probability if they have equal priority for it. Then, it is shown that DA-STB is ex-post constrained efficient, if and only if it is equal-top fair, if and only if the priority structure satisfies a certain acyclic condition. We further characterize the priority structures under which DA-STB is ex-post stable-and-efficient. Based on the characterized priority domains, and using a weak fairness notion called local envy-freeness, new theoretical support is provided for the use of this rule: for any priority structure, among the class of strategy-proof, ex-post stable, symmetric, and locally envy-free rules, each of the above desiderata—ex-post constrained efficiency, ex-post stability-and-efficiency, and equal-top fairness—can be achieved if and only if it can be achieved by DA-STB.

Incomplete Multiview Clustering via Semidiscrete Optimal Transport for Multimedia Data Mining in IoT

With the wide deployment of the Internet of Things (IoT), large volumes of incomplete multiview data that violates data integrity is generated by various applications, which inevitably produces negative impacts on the quality of service of IoT systems. Incomplete multiview clustering (IMC), as an essential technique of data processing, has the potential for mining patterns of incomplete IoT data. However, previous methods utilize notion-strong distances that can only measure differences between distributions at the overlap of data manifolds in fusing complementary information of data for pattern mining. They may suffer from biased estimation and information loss in capturing intrinsic structures of incomplete multiview data. To address these challenges, a semidiscrete multiview optimal transport (SD-MOT) is defined for IMC, which utilizes distances with weak notions to capture intrinsic structures of incomplete multiview data. Specifically, IMC is recast as an equivalent optimal transport between continuous incomplete multiview data and discrete clustering centroids, to avoid the strict assumption on overlap between manifolds in pattern mining. Then, SD-MOT is instantiated as a deep incomplete contrastive clustering network to remedy biased estimation and information loss on intrinsic structures of incomplete multiview data. Afterwards, a variational solution to SD-MOT is derived to effectively train the network parameters for pattern mining. Finally, extensive experiments on four representative incomplete multiview datasets verify the superiority of SD-MOT in comparison with nine baseline methods.

On subsolutions and concavity for fully nonlinear elliptic equations

Abstract Subsolutions and concavity play critical roles in classical solvability, especially a priori estimates, of fully nonlinear elliptic equations. Our first primary goal in this paper is to explore the possibility to weaken the concavity condition. The second is to clarify relations between weak notions of subsolution introduced by Székelyhidi and the author, respectively, in attempt to treat equations on closed manifolds. More precisely, we show that these weak notions of subsolutions are equivalent for equations defined on convex cones of type 1 in the sense defined by Caffarelli, Nirenberg and Spruck.

Time Travelers (and Everyone Else) Cannot Do Otherwise

Many defenders of the possibility of time travel into the past also hold that such time travel places no restrictions on what said time travelers can do. Some hold that it places at least a few restrictions on what time travelers can do. In attempting to resolve this dispute, I reached a contrary conclusion. Time travelers to the past cannot do other than what they in fact do. Using a very weak notion of can, I shall argue that the correspondingly strong cannot do otherwise applies in the case of backwards time travel. I defend this result from objections.

Comment on “An efficient identity-based signature scheme with provable security”

In this comment paper, we reveal a security issue in an efficient identity-based signature (IBS) scheme that has been proven secure in the standard model. More specifically, we perform a key-only attack on the IBS scheme to break its universal unforgeability. Since the security notion of universal unforgeability under key-only attack (uuf-koa) is a weaker notion than the existential unforgeability under chosen message attack (euf-cma), this invalidates the euf-cma security claimed by the IBS scheme. Subsequently, we propose a fix to achieve a stronger seuf-cma security without a random oracle.

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

Weak elastic energy of irregular curves

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our p -energy is defined through a relaxation process, where a suitable p -rotation of inscribed polygons is adopted. The discrete p -rotation we choose has a geometric flavour: a polygon is viewed as an approximation to a smooth curve, and hence its discrete curvature is spread out into a smooth density. For any exponent p greater than 1, the p -energy is finite if and only if the arc-length parametrization of the curve has a second-order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the p th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

Geo-indistinguishable masking: enhancing privacy protection in spatial point mapping

ABSTRACT Spatial point mapping is a useful practice in exploratory point pattern analysis, but it poses significant privacy risks as the identity of individuals may be revealed from the maps. Geomasking methods have been developed to mitigate the risks by displacing spatial points before mapping. However, many of these methods rely on a weak privacy notion called spatial k-anonymity, which is insufficient to withstand the growing amount of spatial data (e.g. land use) that adversaries can use as side information to infer the actual locations of individuals. We proposes a method called geo-indistinguishable masking to address this issue by relying on a strong privacy notion called geo-indistinguishability. This notion ensures consistent levels of privacy protection regardless of any side information. The method consists of two steps. The first step involves creating a masking area for each spatial point to include a set of candidate locations to which the point can be relocated. In the second step, we formulate an optimization model to ensure the masked locations satisfy geo-indistinguishability while minimizing the distance displaced. Computational experiments on a synthetic dataset demonstrate that our proposed method is both efficient and effective in providing strong privacy protection while preserving the spatial point patterns.

P-Like Properties of Meager Ideals and Cardinal Invariants

Abstract We study two particular modifications of the P-property of ideals and related cardinal invariants cof𝒥 (ℐ)and cov+(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on ω. We also provide values of the cardinal invariant cof𝒥 (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin < K 〈𝒜 〉 < K Fin × Fin for any MAD family 𝒜 using the weak P-ideal notion.

Gerrymandering individual fairness

Individual fairness requires that similar individuals are treated similarly. It is supposed to prevent the unfair treatment of individuals on the subgroup level and to overcome the problem that group fairness measures are susceptible to manipulation or gerrymandering. The goal of the present paper is to explore the extent to which individual fairness itself can be gerrymandered. It will be proved that individual fairness can be gerrymandered in the context of predicting scores. Then, it will be argued that individual fairness is a very weak notion of fairness for some choices of feature space and metric. Finally, it will be discussed which properties of (individual) fairness are desirable.

Hypersurfaces with mean curvature prescribed by an ambient function: Compactness results

We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is gν, for a given continuous positive ambient function g, and where ν denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearance of “hidden boundaries”, that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that g does not prescribe the mean curvature in the limit. As a special instance of a more general result, we prove that locally uniform Lq-bounds on the (weakly defined) second fundamental form, for q>1, in addition to the customary locally uniform bounds on the perimeters, lead to a compact class of boundaries with mean curvature prescribed by g.The proof relies on treating the boundaries as oriented integral varifolds, in order to exploit their orientability feature (that is lost when treating them as (unoriented) varifolds). Specifically, it relies on the formulation and analysis of a weak notion of curvature coefficients for oriented integral varifolds (inspired by Hutchinson's work [7]).This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in Lq with q>1 and with mean curvature prescribed by any g∈C0 (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions). Our notions and statements are given in a Riemannian manifold, with the oriented varifolds that need not arise as boundaries (for instance, they could come from two-sided immersions).

Strategyproofness and Proportionality in Party-Approval Multiwinner Elections

In party-approval multiwinner elections the goal is to allocate the seats of a fixed-size committee to parties based on the approval ballots of the voters over the parties. In particular, each voter can approve multiple parties and each party can be assigned multiple seats. Two central requirements in this setting are proportional representation and strategyproofness. Intuitively, proportional representation requires that every sufficiently large group of voters with similar preferences is represented in the committee. Strategyproofness demands that no voter can benefit by misreporting her true preferences. We show that these two axioms are incompatible for anonymous party-approval multiwinner voting rules, thus proving a far-reaching impossibility theorem. The proof of this result is obtained by formulating the problem in propositional logic and then letting a SAT solver show that the formula is unsatisfiable. Additionally, we demonstrate how to circumvent this impossibility by considering a weakening of strategyproofness which requires that only voters who do not approve any elected party cannot manipulate. While most common voting rules fail even this weak notion of strategyproofness, we characterize Chamberlin-Courant approval voting within the class of Thiele rules based on this strategyproofness notion.

A non-smooth Brezis-Oswald uniqueness result

Abstract We classify the non-negative critical points in W 0 1 , p ( Ω ) {W}_{0}^{1,p}\left(\Omega ) of J ( v ) = ∫ Ω H ( D v ) − F ( x , v ) d x , J\left(v)=\mathop{\int }\limits_{\Omega }\hspace{0.15em}H\left(Dv)-F\left(x,v){\rm{d}}x, where H H is convex and positively p p -homogeneous, while t ↦ ∂ t F ( x , t ) / t p − 1 t\hspace{0.33em}\mapsto \hspace{0.33em}{\partial }_{t}F\left(x,t)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{p-1} is non-increasing. Since H H may not be differentiable and F F has a one-sided growth condition, J J is only lower semi-continuous on W 0 1 , p ( Ω ) {W}_{0}^{1,p}\left(\Omega ) . We use a weak notion of critical point for non-smooth functionals, derive sufficient regularity of the latter without an Euler-Lagrange equation available, and focus on the uniqueness part of the results in the study of Brezis and Oswald, using a non-smooth Picone inequality.

In Defense of Evidential Minimalism: Varieties of Criticizability

AbstractThis paper will critically engage with Daniel Buckley's argument against “evidential minimalism” (EM), i.e., the claim that necessarily, bits of evidence (are or) provide epistemic reasons for belief. Buckley argues that in some cases, a subject has strong evidence that p (and fulfills further minimal conditions), does not believe p, but nevertheless is not epistemically criticizable and has no epistemic reason to believe p. I will defend EM by pointing out that Buckley's argument trades on an ambiguity between a strong and a weak notion of criticizability.