Multi-Embedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems

On Metric Ramsey-type Dichotomies

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form "Every finite metric space contains a large subspace that is nearly quilateral or far from being equilateral". We consider two distinct interpretations for being "far from equilateral". Proximity among metric spaces is quantified through the metric distortion D. We provide tight asymptotic answers for these problems. In particular, we show that a phase transition occurs at D=2.Comment: 14 pages, 0 figure

On some low distortion metric Ramsey problems

In this note, we consider the metric Ramsey problem for the normed spaces l_p. Namely, given some 1=1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into l_p with distortion at most alpha. In [arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of $m$ on alpha undergoes a phase transition at alpha=2. Here we consider this problem for other l_p, and specifically the occurrence of a phase transition for p other than 2. It is shown that a phase transition does occur at alpha=2 for every p in the interval [1,2]. For p>2 we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every 1<p<infinity there are arbitrarily large metric spaces, no four points of which embed isometrically in l_p.Comment: 14 pages, to be published in Discrete and Computational Geometr

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio

Effect of geometrical irregularities on propagation delay in axonal trees

Multiple successive geometrical inhomogeneities, such as extensive arborization and terminal varicosities, are usual characteristics of axons. Near such regions the velocity of the action potential (AP) changes. This study uses AXONTREE, a modeling tool developed in the companion paper for two purposes: (a) to gain insights into the consequence of these irregularities for the propagation delay along axons, and (b) to simulate the propagation of APs along a reconstructed axon from a cortical cell, taking into account information concerning the distribution of boutons (release sites) along such axons to estimate the distribution of arrival times of APs to the axons release sites. We used Hodgkin and Huxley (1952) like membrane properties at 20 degrees C. Focusing on the propagation delay which results from geometrical changes along the axon (and not from the actual diameters or length of the axon), the main results are: (a) the propagation delay at a region of a single geometrical change (a step change in axon diameter or a branch point) is in the order of a few tenths of a millisecond. This delay critically depends on the kinetics and the density of the excitable channels; (b) as a general rule, the lag imposed on the AP propagation at a region with a geometrical ratio GR greater than 1 is larger than the lead obtained at a region with a reciprocal of that GR value; (c) when the electronic distance between two successive geometrical changes (Xdis) is small, the delay is not the sum of the individual delays at each geometrical change, when isolated. When both geometrical changes are with GR greater than 1 or both with GR less than 1, this delay is supralinear (larger than the sum of individual delays). The two other combinations yield a sublinear delay; and (d) in a varicose axon, where the diameter changes frequently from thin to thick and back to thin, the propagation velocity may be slower than the velocity along a uniform axon with the thin diameter. Finally, we computed propagation delays along a morphologically characterized axon from layer V of the somatosensory cortex of the cat. This axon projects mainly to area 4 but also sends collaterals to areas 3b and 3a. The model predicts that, for this axon, areas 3a, 3b, and the proximal part of area 4 are activated approximately 2 ms before the activation of the distal part of area 4

Limitations to Frechet's Metric Embedding Method

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur

Ramsey-type theorems for metric spaces with applications to online problems

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary version in FOCS '01. To be published in J. Comput. System Sc

SOX17 is a critical specifier of human primordial germ cell fate.

Specification of primordial germ cells (PGCs) marks the beginning of the totipotent state. However, without a tractable experimental model, the mechanism of human PGC (hPGC) specification remains unclear. Here, we demonstrate specification of hPGC-like cells (hPGCLCs) from germline competent pluripotent stem cells. The characteristics of hPGCLCs are consistent with the embryonic hPGCs and a germline seminoma that share a CD38 cell-surface marker, which collectively defines likely progression of the early human germline. Remarkably, SOX17 is the key regulator of hPGC-like fate, whereas BLIMP1 represses endodermal and other somatic genes during specification of hPGCLCs. Notable mechanistic differences between mouse and human PGC specification could be attributed to their divergent embryonic development and pluripotent states, which might affect other early cell-fate decisions. We have established a foundation for future studies on resetting of the epigenome in hPGCLCs and hPGCs for totipotency and the transmission of genetic and epigenetic information.We thank Rick Livesey and his lab for help with the culture of hESCs; Sohei Kitazawa and Janet Shipley for the TCam-2 cells; Nigel Miller and Andy Riddell for cell sorting, Roger Barker, Xiaoling He, and Pam Tyers for collection of human embryos; and Charles Bradshaw for help with bioinformatics. We thank members of the Surani and Hanna labs for important discussions and technical help. N.I. is supported by Grant-in-Aid for fellows of the JSPS and by BIRAX (the Britain Israel Research and Academic Exchange Partnership) initiative, who provided a project grant to J.H.H. and M.A.S. J.H.H. is supported by Ilana and Pascal Mantoux, the Kimmel Award, ERC (StG-2011-281906), Helmsley Charitable Trust, ISF (Bikura, Morasha, ICORE), ICRF, the Abisch Frenkel Foundation, the Fritz Thyssen Stiftung, Erica and Robert Drake, Benoziyo Endowment fund, and the Flight Attendant Medical Research Institute (FAMRI). J.H.H. is a New York Stem Cell Foundation Robertson Investigator. W.C.C.T. is supported by Croucher Foundation and Cambridge Trust; M.A.S. is supported by HFSP and a Wellcome Trust Investigator Award.This is the final version of the article, originally published in Cell, Volume 160, Issues 1-2, p253–268, 15 January 2015, doi: 10.1016/j.cell.2014.12.01

IPL for DED

Data for IPL for DED due to MGD (FDA study

Construction of an Experimentally-Based Neuronal Network Model to Explore the Function of the Inferior Olive Nucleus

This study is a synthesis between experimental work and theoretical modelling aiming to understand the mechanism of the sub-threshold oscillations (STOs) observed in the neurons of the inferior olive (IO) nucleus. The model was constructed in a bottom-up approach: it started from the construction of a detailed model of physiologically and morphologically reconstructed IO neurons, progressed to a model of electrically coupled passive neurons, advanced to a simplified model of two electrically coupled neurons with active properties, and completed with a large network model of the IO nucleus. The proficiency from level to level was motivated by the requirement to simplify the model of individual IO neurons without loosing essential features. In the first part (chapter 2), IO neurons were stained, their dendritic morphology was reconstructed, and detailed compartmental models were built. It was found that cable properties estimated from these compartmental models were significantly differe..