arXiv subjects

Sakshi Jain

Publications and source records attributed to Sakshi Jain.

18 recordsLinked to original sources

arXiv2026

Shadowing in the presence of singularities: oriented versus standard shadowing, entropy and the structure of recurrent sets

We study two shadowing properties for flows that differ in the allowed reparametrizations of time: oriented shadowing permits arbitrary increasing reparametrizations, whereas standard shadowing requires their distortion to be uniformly close to one. We prove that these notions are distinct already for $C^\infty$ flows on every closed oriented surface. Moreover, such examples are $C^0$-dense among $C^1$ flows with a singularity and consequently, on closed oriented surfaces with non-zero Euler characteristic, they are dense among all $C^1$ flows. We then relate local standard shadowing to recurrence and entropy. A non-trivial chain-transitive set with local standard shadowing forces positive topological entropy unless it is an irreducible almost heteroclinic set. Consequently, for a zero-entropy flow with standard shadowing, every non-trivial chain-recurrent class has this form, and every non-singular one is minimal. For surface flows, we further prove that oriented shadowing together with finitely many singularities forces every chain-recurrent class to be minimal.

math.DS↗

arXiv2026

Annealed Ruelle-Pollicott Resonances

We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.

math.DS↗

arXiv2026

Productionized Fairness Measurement Under Privacy Constraints

Fairness measurements in the form of disaggregated evaluations often rely on demographic signals that are legally constrained or culturally sensitive. Race and ethnicity signals are among the more difficult signals to curate and use for this task. This paper presents Privacy-Preserving Probabilistic Race/Ethnicity Estimation (PPRE) as a method for enabling fairness measurements with respect to race/ethnicity for U.S.\ LinkedIn members in a privacy-preserving manner. PPRE applies privacy technologies (specifically: secure two-party computation, differential privacy, and additive homomorphic encryption) on top of two race/ethnicity demographic signal sources (the Bayesian Improved Surname Geocoding estimator and a sparse golden survey set of self-reported demographics) to power a fairness measurement solution with respect to US-based race/ethnicity demographics. We detail its privacy guarantees and demonstrate its application on candidate- and viewer-side fairness measurements. We close with a transferable framework for institutions seeking to implement similar privacy-preserving measurement infrastructure.

cs.LG↗

arXiv2026

On main eigenvalues of zero-divisor graphs of reduced rings

The problem of characterizing graphs with a prescribed number of main eigenvalues is a long-standing problem in spectral graph theory. Although some constructions are known, only a few produce infinite families of simple connected graphs with exactly $s \ge 2$ main eigenvalues. Zero-divisor graphs form a well-structured class of algebraic graphs whose spectra can be described explicitly using equitable partitions, making them a convenient setting to study main eigenvalues. In this paper, we prove that the zero-divisor graphs of reduced rings provide an infinite family of simple connected graphs with exactly $s$ main eigenvalues, and that certain induced bipartite subgraphs also have exactly $s$ main eigenvalues for any positive integer $s$.

math.CO↗

arXiv2026

Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term

We study stochastic differential equations on the $d$-dimensional flat torus $\mathbb{T}^d$ with drift and perturbation coefficients in $L^{\infty}(\mathbb{T}^d;\mathbb{R}^d)$ and additive non-degenerate noise. For the associated transfer operators, we analyse the dependence of the stationary measure and of the expectation of a given observable on small perturbations of the drift. In this framework, we prove a linear response formula for the invariant density and for the expectation of a given observable. We then address an optimal response problem, namely the determination of admissible perturbations that maximise the first-order variation of a prescribed observable. We establish existence of optimal perturbations and, in a Hilbert space framework, prove uniqueness and provide an explicit characterisation of the optimiser. This yields a practical Fourier-based numerical method, which we implement in several numerical examples, including both low and high-dimensional settings.

math.DS↗

arXiv2025

Embeddings of the line graphs associated with the essential graphs of commutative rings

Let $A$ be a finite commutative ring with unity $1 \neq 0.$ An ideal of $A$ is said to be essential if it has a non-zero intersection with every non-zero ideal of $A.$ The essential graph of $A$ is a simple undirected graph whose vertex set consists of all non-zero zero-divisors of $A.$ Two different vertices $u$ and $v$ are connected by an edge precisely when the ideal formed by the annihilator of their product $uv$ is essential in $A.$ This paper examines the minimal embeddings of the line graph of the essential graph of $A$ into orientable surfaces as well as non-orientable surfaces. Our results include a complete classification of finite commutative rings for which the line graphs of their essential graphs is planar, outerplanar or have genus or crosscap number at most two. We also characterize all such non-local rings for which the line graph of their zero-divisor graph is outerplanar.

math.AC↗

arXiv2025

Optimal response for stochastic differential equations by local kernel perturbations

We consider a random dynamical system on $\mathbb{R}^d$, whose dynamics is defined by a stochastic differential equation. The annealed transfer operator associated with such systems is a kernel operator. Given a set of feasible infinitesimal perturbations $P$ to this kernel, with support in a certain compact set, and a specified observable function $ϕ: \mathbb{R}^d \to \mathbb{R}$, we study which infinitesimal perturbation in $P$ produces the greatest change in expectation of $ϕ$. We establish conditions under which the optimal perturbation uniquely exists and present a numerical method to approximate the optimal infinitesimal kernel perturbation. Finally, we numerically illustrate our findings with concrete examples.

math.DS↗

arXiv2024

Privacy-Preserving Race/Ethnicity Estimation for Algorithmic Bias Measurement in the U.S

AI fairness measurements, including tests for equal treatment, often take the form of disaggregated evaluations of AI systems. Such measurements are an important part of Responsible AI operations. These measurements compare system performance across demographic groups or sub-populations and typically require member-level demographic signals such as gender, race, ethnicity, and location. However, sensitive member-level demographic attributes like race and ethnicity can be challenging to obtain and use due to platform choices, legal constraints, and cultural norms. In this paper, we focus on the task of enabling AI fairness measurements on race/ethnicity for \emph{U.S. LinkedIn members} in a privacy-preserving manner. We present the Privacy-Preserving Probabilistic Race/Ethnicity Estimation (PPRE) method for performing this task. PPRE combines the Bayesian Improved Surname Geocoding (BISG) model, a sparse LinkedIn survey sample of self-reported demographics, and privacy-enhancing technologies like secure two-party computation and differential privacy to enable meaningful fairness measurements while preserving member privacy. We provide details of the PPRE method and its privacy guarantees. We then illustrate sample measurement operations. We conclude with a review of open research and engineering challenges for expanding our privacy-preserving fairness measurement capabilities.

cs.LG↗

arXiv2024

Piecewise Contractions

We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of $\bR^d$. We prove that generically the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.

math.DS↗

arXiv2023

Disentangling and Operationalizing AI Fairness at LinkedIn

Operationalizing AI fairness at LinkedIn's scale is challenging not only because there are multiple mutually incompatible definitions of fairness but also because determining what is fair depends on the specifics and context of the product where AI is deployed. Moreover, AI practitioners need clarity on what fairness expectations need to be addressed at the AI level. In this paper, we present the evolving AI fairness framework used at LinkedIn to address these three challenges. The framework disentangles AI fairness by separating out equal treatment and equitable product expectations. Rather than imposing a trade-off between these two commonly opposing interpretations of fairness, the framework provides clear guidelines for operationalizing equal AI treatment complemented with a product equity strategy. This paper focuses on the equal AI treatment component of LinkedIn's AI fairness framework, shares the principles that support it, and illustrates their application through a case study. We hope this paper will encourage other big tech companies to join us in sharing their approach to operationalizing AI fairness at scale, so that together we can keep advancing this constantly evolving field.

cs.SI↗

arXiv2022

Discontinuities cause essential spectrum

We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds $L^\infty$ and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to the theoretical best possible case.

math.DS↗

arXiv2020

AI Assisted Apparel Design

Fashion is a fast-changing industry where designs are refreshed at large scale every season. Moreover, it faces huge challenge of unsold inventory as not all designs appeal to customers. This puts designers under significant pressure. Firstly, they need to create innumerous fresh designs. Secondly, they need to create designs that appeal to customers. Although we see advancements in approaches to help designers analyzing consumers, often such insights are too many. Creating all possible designs with those insights is time consuming. In this paper, we propose a system of AI assistants that assists designers in their design journey. The proposed system assists designers in analyzing different selling/trending attributes of apparels. We propose two design generation assistants namely Apparel-Style-Merge and Apparel-Style-Transfer. Apparel-Style-Merge generates new designs by combining high level components of apparels whereas Apparel-Style-Transfer generates multiple customization of apparels by applying different styles, colors and patterns. We compose a new dataset, named DeepAttributeStyle, with fine-grained annotation of landmarks of different apparel components such as neck, sleeve etc. The proposed system is evaluated on a user group consisting of people with and without design background. Our evaluation result demonstrates that our approach generates high quality designs that can be easily used in fabrication. Moreover, the suggested designs aid to the designers creativity.

cs.CV↗

arXiv2019

Ergodic Decomposition

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.

math.DS↗

arXiv2017

Subliminal Probing for Private Information via EEG-Based BCI Devices

Martinovic et al. proposed a Brain-Computer-Interface (BCI) -based attack in which an adversary is able to infer private information about a user, such as their bank or area-of-living, by analyzing the user's brain activities. However, a key limitation of the above attack is that it is intrusive, requiring user cooperation, and is thus easily detectable and can be reported to other users. In this paper, we identify and analyze a more serious threat for users of BCI devices. We propose a it subliminal attack in which the victim is attacked at the levels below his cognitive perception. Our attack involves exposing the victim to visual stimuli for a duration of 13.3 milliseconds -- a duration usually not sufficient for conscious perception. The attacker analyzes subliminal brain activity in response to these short visual stimuli to infer private information about the user. If carried out carefully, for example by hiding the visual stimuli within screen content that the user expects to see, the attack may remain undetected. As a consequence, the attacker can scale it to many victims and expose them to the attack for a long time. We experimentally demonstrate the feasibility of our subliminal attack via a proof-of-concept study carried out with 27 subjects. We conducted experiments on users wearing Electroencephalography-based BCI devices, and used portrait pictures of people as visual stimuli which were embedded within the background of an innocuous video for a time duration not exceeding 13.3 milliseconds. Our experimental results show that it is feasible for an attacker to learn relevant private information about the user, such as whether the user knows the identity of the person for which the attacker is probing.

cs.CR↗

arXiv2010

$2N$ qubit "mirror states" for optimal quantum communication

We introduce a new genuinely 2N qubit state, known as the "mirror state" with interesting entanglement properties. The well known Bell and the cluster states form a special case of these "mirror states", for N=1 and N=2 respectively. It can be experimentally realized using $SWAP$ and multiply controlled phase shift operations. After establishing the general conditions for a state to be useful for various communicational protocols involving quantum and classical information, it is shown that the present state can optimally implement algorithms for the quantum teleportation of an arbitrary N qubit state and achieve quantum information splitting in all possible ways. With regard to superdense coding, one can send 2N classical bits by sending only N qubits and consuming N ebits of entanglement. Explicit comparison of the mirror state with the rearranged N Bell pairs and the linear cluster states is considered for these quantum protocols. We also show that mirror states are more robust than the rearranged Bell pairs with respect to a certain class of collisional decoherence.

quant-ph↗

arXiv2010

Splitting of quantum information using N-qubit linear cluster states

We provide a number of schemes for the splitting up of quantum information among $k$ parties using a $N$-qubit linear cluster state as a quantum channel, such that the original information can be reconstructed only if all the parties cooperate. Explicit circuits are provided for these schemes, which are based on the concept of measurement based locking and unlocking of quantum information. These are experimentally feasible as they require measurements to be performed only on product basis.

quant-ph↗

arXiv2009

Secure quantum conversation through non-destructive discrimination of highly entangled multipartite states

"Quantum conversation" is a way in which two parties can communicate classical information with each other using entanglement as a shared resource. We present this scheme using a multipartite entangled state after describing its generation through appropriate circuit diagrams. We make use of a discrimination scheme which allows one to perform a measurement on the system without destroying its entanglement. We later prove that this scheme is secure in a noiseless and a lossless quantum channel.

quant-ph↗

arXiv2009

Non-destructive discrimination of multiparticle cluster states for quantum computation

Discriminating between orthogonal quantum systems without destroying their entanglement is of interest to quantum computation and communication. In this paper, we explicate the schemes for the non-destructive discrimination (NDD) of 16 orthogonal four qubit cluster states and 32 orthogonal five qubit cluster states respectively. This technique will find its applications in various branches of quantum information theory.

quant-ph↗