1,722,645 research outputs found
HERStory Makers 2023: Shubhangi Kandwal
No full textShubhangi Kandwal is a PhD candidate at Trinity College Dublin, University of Dublin, studying biochemistry and drug design. She took part in HERStory Makers 2023.What is HERStory Makers?HERStory Makers is a social media competition for female-identifying early career researchers to share their research, their career journeys, and to inspire the next generation. Winners are selected by public vote. HERStory Makers is also part of EXPLORATHON, Scotland's contribution to European Researchers' Night.In 2022-23, EXPLORATHON was supported by the Engineering & Physical Sciences Research Council [grant number EP/X020762/1].Author contributions to contentShubhangi conceived, planned, and recorded the video content. Kirsty Ross edited the video content to insert HERStory Maker credits and added subtitles.</p
LIPIcs, Volume 169, CCC 2020, Complete Volume
No full textLIPIcs, Volume 169, CCC 2020, Complete Volum
The role of mercy in India
No full textShubhangi Agarwalla argues that the death penalty has bureaucratised death and in doing so, relieved individuals of a sense of their own moral responsibility within its delivery. Considering Derrida's seminal argument on mercy, Indian society must reconsider the pivotal role of mercy within justice, writes Agarwalla
Application of English in Hindi Advertisements
No full textThis is the content and survey data collected and processed during my research on 'Application of English in Hindi Advertisements. ' It provides valuable information on consumer behavior regarding language use in Hindi Television Commercials
Editorial Note
No full textReflections on a year of student scientific research, from Editor-in-Chief Shubhangi Karmakar, and ex. Deputy Editor Lucy Fitzsimmons
Front Matter, Table of Contents, Preface, Conference Organization
No full textFront Matter, Table of Contents, Preface, Conference Organizatio
Evolving Dynamics of Media Policy in India: Cultural and Political Implications of Streaming
No full textArithmetic Circuits with Locally Low Algebraic Rank
No full textIn recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply VP != VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits.
A depth-4 circuit is a representation of an N-variate, degree n polynomial P as P = sum_{i=1}^T Q_{i1} * Q_{i2} * ... * Q_{it} where the Q_{ij} are given by their monomial expansion. Homogeneity adds the constraint that for every i in [T], sum_{j} degree(Q_{ij}) = n. We study an extension where, for every i in [T], the algebraic rank of the set of polynomials {Q_{i1}, Q_{i2}, ... ,Q_{it}} is at most some parameter k. We call this the class of spnew circuits. Already for k=n, these circuits are a strong generalization of the class of homogeneous depth-4 circuits, where in particular t<=n (and hence k<=n).
We study lower bounds and polynomial identity tests for such circuits and prove the following results.
1. Lower bounds: We give an explicit family of polynomials {P_n} of degree n in N = n^{O(1)} variables in VNP, such that any spnewn circuit computing P_n has size at least exp{(Omega(sqrt(n)*log(N)))}. This strengthens and unifies two lines of work: it generalizes the recent exponential lower bounds for homogeneous depth-4 circuits [KLSS14, KS-full] as well as the Jacobian based lower bounds of Agrawal et al. which worked for spnew circuits in the restricted setting where T * k <= n.
2. Hitting sets: Let spnewbounded be the class of spnew circuits with bottom fan-in at most d. We show that if d and k are at most poly(log(N)), then there is an explicit hitting set for spnewbounded circuits of size quasipolynomial in N and the size of the circuit. This strengthens a result of Forbes which showed such quasipolynomial sized hitting sets in the setting where d and t are at most poly(log(N)).
A key technical ingredient of the proofs is a result which states that over any field of characteristic zero (or sufficiently large characteristic), upto a translation, every polynomial in a set of algebraically dependent polynomials can be written as a function of the polynomials in the transcendence basis. We believe this may be of independent interest. We combine this with shifted partial derivative based methods to obtain our final results
Improved Low-Depth Set-Multilinear Circuit Lower Bounds
No full textIn this paper, we prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial f in VNP defined over n² variables, and of degree n, such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). The hard polynomial f comes from the class of Nisan-Wigderson (NW) design-based polynomials.
Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form (log n)^Ω(Δ n^{1/Δ}) was shown for the size of product-depth Δ set-multilinear formulas computing the iterated matrix multiplication (IMM) polynomial of the same degree and over the same number of variables as f. Moreover, our lower bounds are novel for any Δ ≥ 2.
The precise quantitative expression in our lower bound is interesting also because the lower bounds we obtain are "sharp" in the sense that any asymptotic improvement would imply general set-multilinear circuit lower bounds via depth reduction results.
In the setting of general set-multilinear formulas, a lower bound of the form n^Ω(log n) was already obtained by Raz (J. ACM 2009) for the more general model of multilinear formulas. The techniques of LST (which extend the techniques of the same authors in (FOCS 2021)) give a different route to set-multilinear formula lower bounds, and allow them to obtain a lower bound of the form (log n)^Ω(log n) for the size of general set-multilinear formulas computing the IMM polynomial. Our proof techniques are another variation on those of LST, and enable us to show an improved lower bound (matching that of Raz) of the form n^Ω(log n), albeit for the same polynomial f in VNP (the NW polynomial). As observed by LST, if the same n^Ω(log n) size lower bounds for unbounded-depth set-multilinear formulas could be obtained for the IMM polynomial, then using the self-reducibility of IMM and using hardness escalation results, this would imply super-polynomial lower bounds for general algebraic formulas
Near-Optimal Set-Multilinear Formula Lower Bounds
No full textThe seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas.
In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP) defined over Θ(n²) variables and of degree Θ(n), such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n^{Ω(log n)}.
If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022).
Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM 2009).
By known depth-reduction results, our lower bounds are essentially tight for f and in general, for any hard polynomial that is in set-multilinear VBP or set-multilinear VP. Any asymptotic improvement in the lower bound (for a hard polynomial, say, in VNP) would imply super-polynomial lower bounds for general set-multilinear circuits
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