On U-Statistics and Compressed Sensing II: Non-Asymptotic Worst-Case Analysis

In another related work, U-statistics were used for non-asymptotic "average-case" analysis of random compressed sensing matrices. In this companion paper the same analytical tool is adopted differently - here we perform non-asymptotic "worst-case" analysis. Simple union bounds are a natural choice for "worst-case" analyses, however their tightness is an issue (and questioned in previous works). Here we focus on a theoretical U-statistical result, which potentially allows us to prove that these union bounds are tight. To our knowledge, this kind of (powerful) result is completely new in the context of CS. This general result applies to a wide variety of parameters, and is related to (Stein-Chen) Poisson approximation. In this paper, we consider i) restricted isometries, and ii) mutual coherence. For the bounded case, we show that k-th order restricted isometry constants have tight union bounds, when the measurements m = \mathcal{O}(k (1 + \log(n/k))). Here we require the restricted isometries to grow linearly in k, however we conjecture that this result can be improved to allow them to be fixed. Also, we show that mutual coherence (with the standard estimate \sqrt{(4\log n)/m}) have very tight union bounds. For coherence, the normalization complicates general discussion, and we consider only Gaussian and Bernoulli cases here.Comment: 12 pages. Submitted to IEEE Transactions on Signal Processin

The Single-Uniprior Index-Coding Problem: The Single-Sender Case and The Multi-Sender Extension

Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori. In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori, and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This work looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems.Comment: Author final manuscrip

The Multi-Sender Multicast Index Coding

We focus on the following instance of an index coding problem, where a set of receivers are required to decode multiple messages, whilst each knows one of the messages a priori. In particular, here we consider a generalized setting where they are multiple senders, each sender only knows a subset of messages, and all senders are required to collectively transmit the index code. For a single sender, Ong and Ho (ICC, 2012) have established the optimal index codelength, where the lower bound was obtained using a pruning algorithm. In this paper, the pruning algorithm is simplified, and used in conjunction with an appending technique to give a lower bound to the multi-sender case. An upper bound is derived based on network coding. While the two bounds do not match in general, for the special case where no two senders know any message bit in common, the bounds match, giving the optimal index codelength. The results are derived based on graph theory, and are expressed in terms of strongly connected components.Comment: This is an extended version of the same-titled paper accepted and to be presented at the IEEE International Symposium on Information Theory (ISIT), Istanbul, in July 201

Haptic Rendering & Perception Studies for Laparoscopic Surgery Simulation

Los Alamitos, C

Haptic rendering for VR laparoscopic surgery simulation

Adelaide, S

Kondo effect in spin-orbit mesoscopic interferometers

We consider a flux-threaded Aharonov-Bohm ring with an embedded quantum dot coupled to two normal leads. The local Rashba spin-orbit interaction acting on the dot electrons leads to a spin-dependent phase factor in addition to the Aharonov-Bohm phase caused by the external flux. Using the numerical renormalization group method, we find a splitting of the Kondo resonance at the Fermi level which can be compensated by an external magnetic field. To fully understand the nature of this compensation effect, we perform a scaling analysis and derive an expression for the effective magnetic field. The analysis is based on a tight-binding model which leads to an effective Anderson model with a spin-dependent density of states for the transformed lead states. We find that the effective field originates from the combined effect of Rashba interaction and magnetic flux and that it contains important corrections due to electron-electron interactions. We show that the compensating field is an oscillatory function of both the spin-orbit and the Aharonov-Bohm phases. Moreover, the effective field never vanishes due to the particle-hole symmetry breaking independently of the gate voltage.Comment: 9 pages, 5 figure

Star Formation in the Outer Filaments of NGC 1275

We present photometry of the outer star clusters in NGC 1275, the brightest galaxy in the Perseus cluster. The observations were taken using the Hubble Space Telescope Advanced Camera for Surveys. We focus on two stellar regions in the south and south-east, far from the nucleus of the low velocity system (~22 kpc). These regions of extended star formation trace the H alpha filaments, drawn out by rising radio bubbles. In both regions bimodal distributions of colour (B-R)_0 against magnitude are apparent, suggesting two populations of star clusters with different ages; most of the H alpha filaments show no detectable star formation. The younger, bluer population is found to be concentrated along the filaments while the older population is dispersed evenly about the galaxy. We construct colour-magnitude diagrams and derive ages of at most 10^8 years for the younger population, a factor of 10 younger than the young population of star clusters in the inner regions of NGC 1275. We conclude that a formation mechanism or event different to that for the young inner population is needed to explain the outer star clusters and suggest that formation from the filaments, triggered by a buoyant radio bubble either rising above or below these filaments, is the most likely mechanism.Comment: Accepted for publication in MNRAS, 14 pages, 14 figures, 3 table