15 research outputs found
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere
For an -variate order- tensor , define to be the maximum value taken by the
tensor on the unit sphere. It is known that for a random tensor with i.i.d entries, w.h.p. We study the
problem of efficiently certifying upper bounds on via the natural
relaxation from the Sum of Squares (SoS) hierarchy. Our results include:
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies Our upper bound improves a result of Montanari and Richard
(NIPS 2014) when is large.
- We show the above bound is the best possible up to lower order terms,
namely the optimum of the level- SoS relaxation is at least
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies For growing , this improves upon the bound
certified by constant levels of SoS. This answers in part, a question posed by
Hopkins, Shi, and Steurer (COLT 2015), who established the tight
characterization for constant levels of SoS
Van der knaap disease: A case report
Van der Knaap disease or megalencephalic leukoencephalopathy with subcortical cysts (MLC) is a rare autosomal recessive degenerative disorder characterized by megalencephaly, cerebral leukoencephalopathy, and motor deterioration. Most cases reported with this disease in India belong to the Agarwal Community with Consanguinity. Here, we report the case of a 12-year-old boy belonging to this ethnic background presented with a history of delayed motor milestones, ataxia, poor scholastic performance, and seizures. MLC has a benign course and better outcome with life expectancy up to 3rdβ4th decade of life. MLC should be included in differentials of macrocephaly and leukoencephalopathy with characteristic magnetic resonance imaging findings. A precise diagnosis helps for better management and to prognosticate its benign course
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Separating a Voronoi Diagram via Local Search
Given a set P of n points in R^dwe show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of Pwe present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition
Extending Parikh's Theorem to Weighted and Probabilistic Context-Free Grammars
We prove an analog of Parikh's theorem for weighted context-free
grammars over commutative, idempotent semirings, and exhibit a
stochastic context-free grammar with behavior that cannot be realized
by any stochastic right-linear context-free grammar. Finally, we show
that every unary stochastic context-free grammar with
polynomially-bounded ambiguity has an equivalent stochastic
right-linear context-free grammar.Ope