Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes

On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic

Visual perceptual load induces inattentional deafness

In this article, we establish a new phenomenon of “inattentional deafness” and highlight the level of load on visual attention as a critical determinant of this phenomenon. In three experiments, we modified an inattentional blindness paradigm to assess inattentional deafness. Participants made either a low- or high-load visual discrimination concerning a cross shape (respectively, a discrimination of line color or of line length with a subtle length difference). A brief pure tone was presented simultaneously with the visual task display on a final trial. Failures to notice the presence of this tone (i.e., inattentional deafness) reached a rate of 79% in the high-visual-load condition, significantly more than in the low-load condition. These findings establish the phenomenon of inattentional deafness under visual load, thereby extending the load theory of attention (e.g., Lavie, Journal of Experimental Psychology. Human Perception and Performance, 25, 596–616, 1995) to address the cross-modal effects of visual perceptual load

On Two Continuum Armed Bandit Problems in High Dimensions

We consider the problem of continuum armed bandits where the arms are indexed by a compact subset of Rd. For large d, it is well known that mere smoothness assumptions on the reward functions lead to regret bounds that suffer from the curse of dimensionality. A typical way to tackle this in the literature has been to make further assumptions on the structure of reward functions. In this work we assume the reward functions to be intrinsically of low dimension k ≪ d and consider two models: (i) The reward functions depend on only an unknown subset of k coordinate variables and, (ii) a generalization of (i) where the reward functions depend on an unknown k dimensional subspace of Rd. By placing suitable assumptions on the smoothness of the rewards we derive randomized algorithms for both problems that achieve nearly optimal regret bounds in terms of the number of rounds n

Tight Estimates for Eigenvalues of Regular Graphs

It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 \sqrt {d-1} \cos \big({\pi\over 2 k}\big)$. A similar result has been proved by Friedman using more sophisticated tools.</jats:p