arXiv subjects

Leonid Gurvits

Publications and source records attributed to Leonid Gurvits.

At least 19 recordsLinked to original sources

arXiv2026

Unique Minimizers for Permanents, Mixed Discriminants, and Log-concave Polynomials

The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.

math.CO↗

arXiv2024

From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

math.CO↗

arXiv2023

Astrometric Apparent Motion of High-redshift Radio Sources

Radio-loud quasars at high redshift (z > 4) are rare objects in the Universe and rarely observed with Very Long Baseline Interferometry (VLBI). But some of them have flux density sufficiently high for monitoring of their apparent position. The instability of the astrometric positions could be linked to the astrophysical process in the jetted active galactic nuclei in the early Universe. Regular observations of the high-redshift quasars are used for estimating their apparent proper motion over several years. We have undertaken regular VLBI observations of several high-redshift quasars at 2.3 GHz (S band) and 8.4 GHz (X band) with a network of five radio telescopes: 40-m Yebes (Spain), 25-m Sheshan (China), and three 32-m telescopes of the Quasar VLBI Network (Russia) -- Svetloe, Zelenchukskaya, and Badary. Additional facilities joined this network occasionally. The sources have also been observed in three sessions with the European VLBI Network (EVN) in 2018--2019 and one Long Baseline Array (LBA) experiment in 2018. In addition, several experiments conducted with the Very Long Baseline Array (VLBA) in 2017--2018were used to improve the time sampling and the statistics. Based on these 37 astrometric VLBI experiments between 2017 and 2021, we estimated the apparent proper motions of four quasars: 0901+697, 1428+422, 1508+572, and 2101+600.

astro-ph.GA↗

arXiv2022

Capacity Lower Bounds via Productization

We give a sharp lower bound on the capacity of a real stable polynomial, depending only on the value of its gradient at $x = 1$. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in 2000, which was crucial to the analysis of their permanent approximation algorithm. Such inequalities have played an important role in the recent work on operator scaling and its generalizations and applications, and in fact we use our bound to construct a new scaling algorithm for real stable polynomials. In addition, we give a strong improvement on previous lower bounds of the capacity of a non-homogeneous real stable polynomial, depending only on the value of its gradient at $x = 1$. Crucially, this new bound is independent of the degree of the polynomial, and has singly exponential dependence on the number of variables. This compares favorably to the bounds used recently in the fantastic work of Karlin-Klein-Oveis Gharan to give an improved approximation factor for metric TSP, where this dependence is doubly exponential. Such bounds were conjectured to exist by the authors, and thus our new bound should imply further improvement to the approximation factor for metric TSP. The new technique we develop to prove this bound is productization, which says that any real stable polynomial can be approximated at any point in the positive orthant by a product of linear forms. Beyond the results of this paper, our main hope is that this new technique will allow us to avoid "frightening technicalities", in the words of Laurent and Schrijver, that often accompany combinatorial lower bounds.

math.CO↗

arXiv2021

Probing the innermost regions of AGN jets and their magnetic fields with RadioAstron. V. Space and ground millimeter-VLBI imaging of OJ 287

We present the first polarimetric space VLBI observations of OJ 287, observed with RadioAstron at 22 GHz during a perigee session on 2014 April 4 and five near-in-time snapshots, together with contemporaneous ground VLBI observations at 15, 43, and 86 GHz. Ground-space fringes were obtained up to a projected baseline of 3.9 Earth diameters during the perigee session, and at a record 15.1 Earth diameters during the snapshot sessions, allowing us to image the innermost jet at an angular resolution of $\sim50μ$as, the highest ever achieved at 22 GHz for OJ 287. Comparison with ground-based VLBI observations reveals a progressive jet bending with increasing angular resolution that agrees with predictions from a supermassive binary black hole model, although other models cannot be ruled out. Spectral analyses suggest that the VLBI core is dominated by the internal energy of the emitting particles during the onset of a multi-wavelength flare, while the parsec-scale jet is consistent with being in equipartition between the particles and magnetic field. Estimated minimum brightness temperatures from the visibility amplitudes show a continued rising trend with projected baseline length up to $10^{13}$ K, reconciled with the inverse Compton limit through Doppler boosting for a jet closely oriented to the line of sight. The observed electric vector position angle suggests that the innermost jet has a predominantly toroidal magnetic field, which together with marginal evidence of a gradient in rotation measure across the jet width indicate that the VLBI core is threaded by a helical magnetic field, in agreement with jet formation models.

astro-ph.HE↗

arXiv2020

Counting Matchings via Capacity Preserving Operators

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilized to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g., the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland's lower matching conjecture.

math.CO↗

arXiv2020

VLBI20-30: a scientific roadmap for the next decade -- The future of the European VLBI Network

This white paper describes the science case for Very Long Baseline Interferometry (VLBI) and provides suggestions towards upgrade paths for the European VLBI Network (EVN). The EVN is a distributed long-baseline radio interferometric array, that operates at the very forefront of astronomical research. Recent results, together with the new science possibilities outlined in this vision document, demonstrate the EVN's potential to generate new and exciting results that will transform our view of the cosmos. Together with e-MERLIN, the EVN provides a range of baseline lengths that permit unique studies of faint radio sources to be made over a wide range of spatial scales. The science cases are reviewed in six chapters that cover the following broad areas: cosmology, galaxy formation and evolution, innermost regions of active galactic nuclei, explosive phenomena and transients, stars and stellar masers in the Milky Way, celestial reference frames and space applications. The document concludes with identifying the synergies with other radio, as well as multi-band/multi-messenger instruments, and provide the recommendations for future improvements. The appendices briefly describe other radio VLBI arrays, the technological framework for EVN developments, and a selection of spectral lines of astrophysical interest below 100 GHz. The document includes a glossary for non-specialists, and a list of acronyms at the end.

astro-ph.IM↗

arXiv2020

Complexity of full counting statistics of free quantum particles in product states

We study the computational complexity of quantum-mechanical expectation values of single-particle operators in bosonic and fermionic multi-particle product states. Such expectation values appear, in particular, in full-counting-statistics problems. Depending on the initial multi-particle product state, the expectation values may be either easy to compute (the required number of operations scales polynomially with the particle number) or hard to compute (at least as hard as a permanent of a matrix). However, if we only consider full counting statistics in a finite number of final single-particle states, then the full-counting-statistics generating function becomes easy to compute in all the analyzed cases. We prove the latter statement for the general case of the fermionic product state and for the single-boson product state (the same as used in the boson-sampling proposal). This result may be relevant for using multi-particle product states as a resource for quantum computing.

quant-ph↗

arXiv2019

The radioscience LaRa instrument onboard ExoMars 2020 to investigate the rotation and interior of Mars

LaRa (Lander Radioscience) is an experiment on the ExoMars 2020 mission that uses the Doppler shift on the radio link due to the motion of the ExoMars platform tied to the surface of Mars with respect to the Earth ground stations (e.g. the deep space network stations of NASA), in order to precisely measure the relative velocity of the lander on Mars with respect to the Earth. The LaRa measurements shall improve the understanding of the structure and processes in the deep interior of Mars by obtaining the rotation and orientation of Mars with a better precision compared to the previous missions. In this paper, we provide the analysis done until now for the best realization of these objectives. We explain the geophysical observation that will be reached with LaRa (Length-of-day variations, precession, nutation, and possibly polar motion). We develop the experiment set up, which includes the ground stations on Earth (so-called ground segment). We describe the instrument, i.e. the transponder and its three antennas. We further detail the link budget and the expected noise level that will be reached. Finally, we detail the expected results, which encompasses the explanation of how we shall determine Mars' orientation parameters, and the way we shall deduce Mars' interior structure and Mars' atmosphere from them. Lastly, we explain briefly how we will be able to determine the Surface platform position.

astro-ph.EP↗

arXiv2019

Peering into the Dark (Ages) with Low-Frequency Space Interferometers

Neutral hydrogen pervades the infant Universe, and its redshifted 21-cm signal allows one to chart the Universe. This signal allows one to probe astrophysical processes such as the formation of the first stars, galaxies, (super)massive black holes and enrichment of the pristine gas from z~6 to z~30, as well as fundamental physics related to gravity, dark matter, dark energy and particle physics at redshifts beyond that. As one enters the Dark Ages (z>30), the Universe becomes pristine. Ground-based low-frequency radio telescopes aim to detect the spatial fluctuations of the 21-cm signal. Complementary, global 21-cm experiments aim to measure the sky-averaged 21-cm signal. Escaping RFI and the ionosphere has motivated space-based missions, such as the Dutch-Chinese NCLE instrument (currently in lunar L2), the proposed US-driven lunar or space-based instruments DAPPER and FARSIDE, the lunar-orbit interferometer DSL (China), and PRATUSH (India). To push beyond the current z~25 frontier, though, and measure both the global and spatial fluctuations (power-spectra/tomography) of the 21-cm signal, low-frequency (1-100MHz; BW~50MHz; z>13) space-based interferometers with vast scalable collecting areas (1-10-100 km2), large filling factors (~1) and large fields-of-view (4pi sr.) are needed over a mission lifetime of >5 years. In this ESA White Paper, we argue for the development of new technologies enabling interferometers to be deployed, in space (e.g. Earth-Sun L2) or in the lunar vicinity (e.g. surface, orbit or Earth-Moon L2), to target this 21-cm signal. This places them in a stable environment beyond the reach of most RFI from Earth and its ionospheric corruptions, enabling them to probe the Dark Ages as well as the Cosmic Dawn, and allowing one to investigate new (astro)physics that is inaccessible in any other way in the coming decades. [Abridged]

astro-ph.IM↗

arXiv2019

Discovering the Sky at the Longest Wavelengths with Small Satellite Constellations

Due to ionosphere absorption and the interference by natural and artificial radio emissions, ground observation of the sky at the decameter or longer is very difficult. This unexplored part of electromagnetic spectrum has the potential of great discoveries, notably in the study of cosmic dark ages and dawn, but also in heliophysics and space weather, planets, cosmic ray and neutrinos, pulsar and interstellar medium, extragalactic radio sources, and even SETI. At a forum organized by the International Space Science Institute-Beijing (ISSI-BJ), we discussed the prospect of opening up this window for astronomical observations by using a constellation of small or micro-satellites. We discussed the past experiments and the current ones such as the low frequency payload on Chang'e-4 mission lander, relay satellite and the Longjiang satellite, and also the future DSL mission, which is a linear array on lunar orbit which can make synthesized map of the whole sky as well as measure the global spectrum. We also discuss the synergy with other experiments, including ground global experiments such as EDGES, SARAS, SCI-HI and High-z, PRIZM/Albatros, ground imaging facillities such as LOFAR and MWA, and space experiments such as SUNRISE, DARE/DAPPER and PRATUSH. We also discussed some technical aspects of the DSL concept.

astro-ph.IM↗

arXiv2019

Operator scaling: theory and applications

In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over $\mathbb{Q}$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (whether or not randomization is allowed). The algorithm efficiently solves the "word problem" for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential-time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits, who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp-Lieb inequalities). Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections.

cs.CC↗

arXiv2018

Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling

The celebrated Brascamp-Lieb (BL) inequalities (and their extensions) are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory. While their structural theory is very well understood, far less is known about computing their main parameters. We give polynomial time algorithms to compute feasibility of BL-datum, the optimal BL-constant and a weak separation oracle for the BL-polytope. The same result holds for the so-called Reverse BL inequalities of Barthe. The best known algorithms for any of these tasks required at least exponential time. The algorithms are obtained by a simple efficient reduction of a given BL-datum to an instance of the Operator Scaling problem defined by Gurvits, for which the present authors have provided a polynomial time algorithm. This reduction implies algorithmic versions of many of the known structural results, and in some cases provide proofs that are different or simpler than existing ones. Of particular interest is the fact that the operator scaling algorithm is continuous in its input. Thus as a simple corollary of our reduction we obtain explicit bounds on the magnitude and continuity of the BL-constant in terms of the BL-data. To the best of our knowledge no such bounds were known, as past arguments relied on compactness. The continuity of BL-constants is important for developing non-linear BL inequalities that have recently found so many applications.

cs.CC↗

arXiv2017

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{γ+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.

math.CO↗

arXiv2015

Space-time Dynamics Estimation from Space Mission Tracking Data

Many physical parameters that can be estimated from space mission tracking data influence both the translational dynamics and proper time rates of observers. These different proper time rates cause a variability of the time transfer observable beyond that caused by their translational (and rotational) dynamics. With the near-future implementation of (interplanetary) transponder laser ranging, these effects will become increasingly important, requiring a re-evaluation of the common data analysis practice of using a priori time ephemerides, which is the goal of this paper. We develop a framework for the simultaneous estimation of the initial translational state and the initial proper time of an observer, with the goal of facilitating robust tracking data analysis from next-generation space missions carrying highly accurate clocks and tracking equipment. Using our approach, the influence of physical parameters on both translational and time dynamics are considered at the same level in the analysis, and mutual correlations between the signatures of the two are automatically identified. We perform a covariance analysis using our proposed method with simulated laser data from Earth-based stations to both a Mars and Mercury lander. Using four years of tracking data for the Mars lander simulations, we find a difference between our results using the simultaneous space-time dynamics estimation and the classical analysis technique (with an ta priori time ephemeris) of around 0.1 % in formal errors and correlation coefficients. For a Mercury lander this rises to around 1% for a 1-month mission and 10 % for a 4-year mission. By means of Monte Carlo simulation, we find that using an a priori time ephemeris of representative accuracy will result in estimation errors that are orders of magnitude above the formal error when processing highly accurate laser time transfer data.

astro-ph.EP↗

arXiv2015

Studies of Relativistic Jets in Active Galactic Nuclei with SKA

Relativistic jets in active galactic nuclei (AGN) are among the most powerful astrophysical objects discovered to date. Indeed, jetted AGN studies have been considered a prominent science case for SKA, and were included in several different chapters of the previous SKA Science Book (Carilli & Rawlings 2004). Most of the fundamental questions about the physics of relativistic jets still remain unanswered, and await high-sensitivity radio instruments such as SKA to solve them. These questions will be addressed specially through analysis of the massive data sets arising from the deep, all-sky surveys (both total and polarimetric flux) from SKA1. Wide-field very-long-baseline-interferometric survey observations involving SKA1 will serve as a unique tool for distinguishing between extragalactic relativistic jets and star forming galaxies via brightness temperature measurements. Subsequent SKA1 studies of relativistic jets at different resolutions will allow for unprecedented cosmological studies of AGN jets up to the epoch of re-ionization, enabling detailed characterization of the jet composition, magnetic field, particle populations, and plasma properties on all scales. SKA will enable us to study the dependence of jet power and star formation on other properties of the AGN system. SKA1 will enable such studies for large samples of jets, while VLBI observations involving SKA1 will provide the sensitivity for pc-scale imaging, and SKA2 (with its extraordinary sensitivity and dynamic range) will allow us for the first time to resolve and model the weakest radio structures in the most powerful radio-loud AGN.

astro-ph.GA↗

arXiv2014

Very Long Baseline Interferometry with the SKA

Adding VLBI capability to the SKA arrays will greatly broaden the science of the SKA, and is feasible within the current specifications. SKA-VLBI can be initially implemented by providing phased-array outputs for SKA1-MID and SKA1-SUR and using these extremely sensitive stations with other radio telescopes, and in SKA2 by realising a distributed configuration providing baselines up to thousands of km, merging it with existing VLBI networks. The motivation for and the possible realization of SKA-VLBI is described in this paper.

astro-ph.IM↗

arXiv2014

Searching for Extraterrestrial Intelligence with the Square Kilometre Array

The vast collecting area of the Square Kilometre Array (SKA), harnessed by sensitive receivers, flexible digital electronics and increased computational capacity, could permit the most sensitive and exhaustive search for technologically-produced radio emission from advanced extraterrestrial intelligence (SETI) ever performed. For example, SKA1-MID will be capable of detecting a source roughly analogous to terrestrial high-power radars (e.g. air route surveillance or ballistic missile warning radars, EIRP (EIRP = equivalent isotropic radiated power, ~10^17 erg sec^-1) at 10 pc in less than 15 minutes, and with a modest four beam SETI observing system could, in one minute, search every star in the primary beam out to ~100 pc for radio emission comparable to that emitted by the Arecibo Planetary Radar (EIRP ~2 x 10^20 erg sec^-1). The flexibility of the signal detection systems used for SETI searches with the SKA will allow new algorithms to be employed that will provide sensitivity to a much wider variety of signal types than previously searched for. Here we discuss the astrobiological and astrophysical motivations for radio SETI and describe how the technical capabilities of the SKA will explore the radio SETI parameter space. We detail several conceivable SETI experimental programs on all components of SKA1, including commensal, primary-user, targeted and survey programs and project the enhancements to them possible with SKA2. We also discuss target selection criteria for these programs, and in the case of commensal observing, how the varied use cases of other primary observers can be used to full advantage for SETI.

astro-ph.IM↗