Professor
Computer Science
University of Washington
574 Paul G. Allen Center
jrl [at] cs washington edu
Teaching:
SP 21: The art and science of positive definite matricesStudents:
Farzam Ebrahimnejad (co-advised with Shayan)
Research interests:
Algorithms, complexity, and the theory of computation. Geometry and analysis at the interface between the continuous and discrete.
Probability and stochastic processes.
Metric embeddings, spectral graph theory, convex optimization.
[CV]
Teaching:
SP 21: The art and science of positive definite matricesStudents:
Farzam Ebrahimnejad (co-advised with Shayan)
Research interests:
Algorithms, complexity, and the theory of computation. Geometry and analysis at the interface between the continuous and discrete.
Probability and stochastic processes.
Metric embeddings, spectral graph theory, convex optimization.
There is a randomized online algorithm for metrical task systems (MTS) that is \(1\)-competitive for service costs and \(O(\log n)\)-competitive for movement costs. In general, these refined guarantees are optimal up to the implicit constant. While an \(O(\log n)\)-competitive algorithm for MTS on HST metrics was developed in BCLL19, that approach could only establish an \(O((\log n)^2)\)-competitive ratio when the service costs are required to be \(O(1)\)-competitive. Our algorithm is an instantiation of online mirror descent with the regularizer derived from a multiscale conditional entropy.
In fact, our algorithm satisfies a set of even more refined guarantees; we are able to exploit this property to combine it with known random embedding theorems and obtain, for any \(n\)-point metric space, a randomized algorithm that is \(1\)-competitive for service costs and \(O((\log n)^2)\)-competitive for movement costs.
@inproceedings{CL19, AUTHOR = {Christian Coester and James R. Lee}, BOOKTITLE = {Conference on Learning Theory, {COLT} 2019, 25-28 June 2019, Phoenix, AZ, {USA}}, PAGES = {835--848}, TITLE = {Pure entropic regularization for metrical task systems}, YEAR = {2019} }
We consider metrical task systems on tree metrics, and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework. For the special case of HSTs, we refine the standard approach based on combining unfair metrical task systems, yielding an $O(\log n)$-competitive algorithm. This is tight up to the implicit constant factor. Using known random embeddings of finite metric spaces into HSTs, this yields an $O((\log n)^2)$-competitive algorithm for general $n$-point spaces.
@inproceedings{BCLL19, AUTHOR = {Bubeck, S\'{e}bastien and Cohen, Michael B. and Lee, James R. and Lee, Yin Tat}, BOOKTITLE = {Proceedings of the {T}hirtieth {A}nnual {ACM}-{SIAM} {S}ymposium on {D}iscrete {A}lgorithms}, PAGES = {89--97}, PUBLISHER = {SIAM, Philadelphia, PA}, TITLE = {Metrical task systems on trees via mirror descent and unfair gluing}, YEAR = {2019} }
We establish new and improved bounds on the flow-cut gap in planar flow networks where the terminals lie on a small number of faces. The classical Okamura-Seymour theorem (1981) shows that when all terminals lie on a single face, the flow-cut gap is exactly 1. We show that if the terminals lie on $\gamma$ faces, the flow-cut gap is $O(\log \gamma)$. This is proved by embedding the terminal metric into a distribution over trees with distortion $O(\log \gamma)$, which is tight up to the implied constant factor.
@inproceedings{KLR19, AUTHOR = {Krauthgamer, Robert and Lee, James R. and Rika, Havana}, BOOKTITLE = {Proceedings of the {T}hirtieth {A}nnual {ACM}-{SIAM} {S}ymposium on {D}iscrete {A}lgorithms}, PAGES = {525--534}, PUBLISHER = {SIAM, Philadelphia, PA}, TITLE = {Flow-cut gaps and face covers in planar graphs}, YEAR = {2019} }
We give a poly(log k)-competitive randomized algorithm for the k-server problem on general metric spaces. The best previous result independent of the underlying metric space is the 2k-1 competitive ratio established for the deterministic work function algorithm by Koutsoupias and Papadimitriou (1995).
Since deterministic algorithms can do no better than k on any metric space with at least k+1 points, this establishes that for every metric space on which the problem is non-trivial, randomized algorithms give an exponential improvement over deterministic algorithms. The approach employs our previous algorithm for the fractional k-server problem on HSTs.
@incollection{LeeFOCS18, AUTHOR = {Lee, James R.}, BOOKTITLE = {59th {A}nnual {IEEE} {S}ymposium on {F}oundations of {C}omputer {S}cience---{FOCS} 2018}, PAGES = {438--449}, PUBLISHER = {IEEE Computer Soc., Los Alamitos, CA}, TITLE = {Fusible {HST}s and the randomized {$k$}-server conjecture}, YEAR = {2018} }
We present an \(O((\log k)^2)\)-competitive randomized algorithm for the k-server problem on hierarchically separated trees (HSTs). This is the first o(k)-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy.
When combined with Bartal’s static HST embedding reduction, this leads to an \(O((\log k)^2 \log n)\)-competitive algorithm on any n-point metric space. We give a new dynamic HST embedding that yields an \(O((\log k)^3 \log \Delta)\)-competitive algorithm on any metric space where the ratio of the largest to smallest non-zero distance is at most \(\Delta\).
@inproceedings{BCLLM18, author = {S{\'{e}}bastien Bubeck and Michael B. Cohen and Yin Tat Lee and James R. Lee and Aleksander Madry}, title = {k-server via multiscale entropic regularization}, booktitle = {Proceedings of the 50th Annual {ACM} {SIGACT} Symposium on Theory of Computing, {STOC} 2018, Los Angeles, CA, USA, June 25-29, 2018}, pages = {3--16}, year = {2018}, }
A region intersection graph over a base graph \(G_0\) is a graph \(G\) whose vertices correspond to connected subsets of \(G_0\) with an edge between two vertices if the corresponding regions intersect. We show that if \(G_0\) excludes the complete graph \(K_h\) as a minor, then every region intersection graph over \(G_0\) with m edges has a balanced separator with \(O(\sqrt{m})\) nodes. If \(G\) has uniformly bounded vertex degrees, we show the separator is found by spectral partitioning.
A string graph is the intersection graph of continuous arcs in the plane. The preceding result implies that every string graph with m edges has a balanced separator of size \(O(\sqrt{m})\). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010) and improves over the \(O(\sqrt{m} \log m)\) bound of Matousek (2013).
@inproceedings{LeeStrings17, author = {James R. Lee}, title = {Separators in Region Intersection Graphs}, booktitle = {8th Innovations in Theoretical Computer Science Conference, {ITCS} 2017, January 9-11, 2017, Berkeley, CA, {USA}}, pages = {1:1--1:8}, year = {2017}, }
We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents:
\[\begin{align*} d_w &= d_f + \tilde{\zeta} \\ d_s &= 2d_f/d_w, \end{align*}\]where $d_w$ is the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral dimension, and $\tilde{\zeta}$ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if $d_f$ and $\tilde{\zeta} \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned equalities hold. Moreover, our primary new estimate $d_w \geq d_f + \tilde{\zeta}$ is established for all $\tilde{\zeta} \in \mathbb{R}$.
This has a number of consequences for unimodular random networks arising from statistical physics.
We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinisc graph metric). Kesten (1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger.
More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multiscale way of covering the graph so that “deep patches” have “thin backbones.” Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents.
Consider an infinite planar graph with uniform polynomial growth of degree $d > 2$. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of counterexamples. In particular, we show that for every rational $d > 2$, there is a planar graph with uniform polynomial growth of degree $d$ on which the random walk is transient, disproving a conjecture of Benjamini (2011).
By a well-known theorem of Benjamini and Schramm, such a graph cannot be a unimodular random graph. We also give examples of unimodular random planar graphs of uniform polynomial growth with unexpected properties. For instance, graphs of (almost sure) uniform polynomial growth of every rational degree $d > 2$ for which the speed exponent of the walk is larger than $1/d$, and in which the complements of all balls are connected. This resolves negatively two questions of Benjamini and Papasoglou (2011).
If \((G, \rho)\) is the distributional limit of finite graphs that can be sphere-packed in \(\mathbb{R}^d\), then the conformal growth exponent of \((G, \rho)\) is at most d. In other words, there exists a unimodular “unit volume” weighting of the graph metric on G such that the volume growth of balls is asymptotically polynomial with exponent d. This generalizes to graphs that can be quasisymmetrically packed in an Ahlfors d-regular metric measure space. Using our previous work, it gives bounds on the almost sure spectral dimension of G.
@article{LeeGAFA18, AUTHOR = {Lee, James R.}, FJOURNAL = {Geometric and Functional Analysis}, JOURNAL = {Geom. Funct. Anal.}, NUMBER = {4}, PAGES = {1091--1130}, TITLE = {Discrete uniformizing metrics on distributional limits of sphere packings}, VOLUME = {28}, YEAR = {2018} }
[credit: Jérémie Bettinelli]
For a unimodular random graph \((G, \rho)\), we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of \((G, \rho)\), which is the best asymptotic degree of volume growth of balls that can be achieved by such a weighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension.
For two-dimensional growth, one obtains more precise information about recurrence, the heat kernel, and subdiffusivity of the random walk. In particular, this gives a new method for studying the uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ).
We show that if $(G,\rho)$ is a stationary random graph of annealed polynomial growth, then almost surely there is an infinite sequence of times at which the random walk started from x is at most diffusive. This result is new even in the case when G is a stationary random subgraph of $\mathbb{Z}^d$. Combined with the work of Benjamini, Duminil-Copin, Kozma, and Yadin, it implies that G almost surely does not admit a non-constant harmonic function of sublinear growth.
To complement this, we argue that passing to a subsequence of times is necessary, as there are stationary random graphs of (almost sure) polynomial growth where the random walk is almost surely superdiffisuve at an infinite subset of times.
Addendum: In the paper, we ask whether passing to a subsequence of times is necessary when $G$ is a stationary random subgraph of $\mathbb{Z}^d$. For the case $d=2$, it holds for all times using my paper with Ding and Peres. For $d$ sufficiently large, passing to a subsequence of times is necessary; this can be proved by showing that our stationary random graph occurs as a stationary random subgraph of $\mathbb{Z}^d_{\infty}$, using my paper with Krauthgamer.
@article {GLP17, AUTHOR = {Ganguly, Shirshendu and Lee, James R. and Peres, Yuval}, TITLE = {Diffusive estimates for random walks on stationary random graphs of polynomial growth}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {27}, YEAR = {2017}, NUMBER = {3}, PAGES = {596--630}, }
[credit: Bernd Sturmfels]
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than \(2^{n^c}\), for some constant \(c > 0\). This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT.
[ PPT slides ]
Best paper award, STOC 2015
@inproceedings{LRS15, AUTHOR = {James R. Lee and Prasad Raghavendra and David Steurer}, BOOKTITLE = {Proceedings of the Forty-Seventh Annual {ACM} on Symposium on Theory of Computing, {STOC} 2015, Portland, OR, USA, June 14-17, 2015}, NOTE = {Full version at \verb|https://arxiv.org/abs/1411.6317|}, PAGES = {567--576}, TITLE = {Lower Bounds on the Size of Semidefinite Programming Relaxations}, YEAR = {2015} }
We show that under the Ornstein-Uhlenbeck semigroup (i.e., the natural diffusion process) on n-dimensional Gaussian space, any nonnegative, measurable function exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989).
Video: Talagrand’s convolution conjecture and geometry via coupling (IAS)
@article{ElDuke18, AUTHOR = {Eldan, Ronen and Lee, James R.}, FJOURNAL = {Duke Mathematical Journal}, JOURNAL = {Duke Math. J.}, NUMBER = {5}, PAGES = {969--993}, TITLE = {Regularization under diffusion and anticoncentration of the information content}, VOLUME = {167}, YEAR = {2018} }
[credit: Fiorini, Rothvoss, and Tiwary]
We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8.
Video: Linear programming and constraint satisfaction (Simons Institute)
@article {CLRS16, AUTHOR = {Chan, Siu On and Lee, James R. and Raghavendra, Prasad and Steurer, David}, TITLE = {Approximate constraint satisfaction requires large {LP} relaxations}, JOURNAL = {J. ACM}, FJOURNAL = {Journal of the ACM}, VOLUME = {63}, YEAR = {2016}, NUMBER = {4}, PAGES = {Art. 34, 22}, }
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger’s inequality and its variants provide a robust version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.
It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into \(\mathbb{R}^k\), and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of measure ≈ 1/k and the kth smallest eigenvalue of the normalized Laplacian.
Notes: A no frills proof of the higher-order Cheeger inequality
Related: One hundred hours of lectures from the SGT program at the Simons Institute.
Related: Laurent Miclo uses the higher-order Cheeger inequality for the basis of his resolution of Hoegh-Krohn and Simon’s conjecture that every hyperbounded operator has a spectral gap.
@article{LOT14, AUTHOR = {Lee, James R. and Gharan, Shayan Oveis and Trevisan, Luca}, FJOURNAL = {Journal of the ACM}, JOURNAL = {J. ACM}, NUMBER = {6}, PAGES = {Art. 37, 30}, TITLE = {Multiway spectral partitioning and higher-order {C}heeger inequalities}, VOLUME = {61}, YEAR = {2014} }
We show that the cover time of a graph can be related to the square of the maximum of the associated Gaussian free field. This yields a positive answer to a question of Aldous and Fill (1994) on deterministic approximations to the cover time, and positively resolves the Blanket Time conjecture of Winkler and Zuckerman (1996).
Video: Cover times of graphs and the Gaussian free field (Newton Institute)
Notes: Majorizing measures and Gaussian processes
Related questions and conjectures (all solved except the one after Lemma 4)
See also the related preprint of Alex Zhai that resolves our main conjecture.
@article{DLP12, AUTHOR = {Ding, Jian and Lee, James R. and Peres, Yuval}, FJOURNAL = {Annals of Mathematics. Second Series}, JOURNAL = {Ann. of Math. (2)}, NUMBER = {3}, PAGES = {1409--1471}, TITLE = {Cover times, blanket times, and majorizing measures}, VOLUME = {175}, YEAR = {2012} }
A method is presented for establishing an upper bound on the first non-trivial eigenvalue of the Laplacian of a finite graph. Our approach uses multi-commodity flows to deform the geometry of the graph, and the resulting metric is embedded into Euclidean space to recover a bound on the Rayleigh quotient.
Using this, we resolve positively a question of Spielman and Teng by proving that \(\lambda_2(G) \leq O(d h^6 \log h/n)\) whenever \(G\) is a \(K_h\)-minor-free graph with maximum degree \(d\). While the standard “sweep” algorithm applied to the second eigenvector of a graph my fail to find a good quotient cut in graphs of unbounded degree, the arguments here produce a vector that works for arbitrary graphs. This yields an alterante proof of the Alon-Seymour-Thomas theorem that every excluded-minor family of graphs has \(O(\sqrt{n})\)-node balanced separators.
Related: These methods are extended to higher eigenvalues in a
joint work with Kelner, Price, and Teng.
Related: These method form the basis for the uniformization and heat kernel bounds in Conformal
growth rates and spectral geometry on distributional limits of graphs.
@article {MR2665882, AUTHOR = {Biswal, Punyashloka and Lee, James R. and Rao, Satish}, TITLE = {Eigenvalue bounds, spectral partitioning, and metrical deformations via flows}, JOURNAL = {J. ACM}, FJOURNAL = {Journal of the ACM}, VOLUME = {57}, YEAR = {2010}, NUMBER = {3}, PAGES = {Art. 13, 23}, }
[credit: Patrick Massot]
We prove that the function \(d : \mathbb{R}^3 \times \mathbb{R}^3 \to [0,\infty)\) given by
is a metric on \(\mathbb{R}^3\) such that \((\mathbb{R}^3, \sqrt{d})\) is isometric to a subset of Hilbert space, yet \((\mathbb{R}^3,d)\) does not admit a bi-Lipschitz embedding into \(L_1\). This yields a new simple counter example to the Goemans-Linial conjecture on the integrality gap of the SDP relaxation of the Sparsest Cut problem.
Our methods involve Fourier analytic techniques, combined with a breakthrough of Cheeger and Kleiner, along with classical results of Pansu on the differentiability of Lispchitz functions on the Heisenberg group.
@inproceedings{LN06, author = {James R. Lee and Assaf Naor}, title = {Lp metrics on the Heisenberg group and the Goemans-Linial conjecture}, booktitle = {47th Annual {IEEE} Symposium on Foundations of Computer Science {(FOCS} 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings}, pages = {99--108}, year = {2006}, }
We prove that every \(n\)-point metric space of negative type (and, in particular, every \(n\)-point subset of \(L_1\)) embeds into a Euclidean space with distortion \(O(\sqrt{\log n} \log \log n)\), a result which is tight up to the iterated logarithm factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. If the demand is supported on a set of size \(k\), we achieve an approximation ratio of \(O(\sqrt{\log k} \log \log k)\).
@article {LN08, AUTHOR = {Arora, Sanjeev and Lee, James R. and Naor, Assaf}, TITLE = {Euclidean distortion and the sparsest cut}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {21}, YEAR = {2008}, NUMBER = {1}, PAGES = {1--21}, }
My research has been generously supported by the National Science Foundation, the Simons Foundation, the Sloan Foundation, and Microsoft.