Algorithmic linear dimension reduction in the l_1 norm for sparse vectors

This paper develops a new method for recovering m-sparse signals that is simultaneously uniform and quick. We present a reconstruction algorithm whose run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal. The reconstruction error is within a logarithmic factor (in m) of the optimal m-term approximation error in l_1. In particular, the algorithm recovers m-sparse signals perfectly and noisy signals are recovered with polylogarithmic distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a logarithmic factor of optimal. We also present a small-space implementation of the algorithm. These sketching techniques and the corresponding reconstruction algorithms provide an algorithmic dimension reduction in the l_1 norm. In particular, vectors of support m in dimension d can be linearly embedded into O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)). Furthermore, this reconstruction is stable and robust under small perturbations

A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization

We combine resolvent-mode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, non-empirical, and scalable with Reynolds number. To optimally represent DNS data at friction Reynolds number $2003$, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only $12$ modes per wall-parallel wavenumber pair and temporal frequency, we obtain close agreement with DNS-spectra, reducing the wall-normal and temporal resolutions used in the simulation by three orders of magnitude

Constructing packings in Grassmannian manifolds via alternating projection

This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini--Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.Comment: 41 pages, 7 tables, 4 figure

Cultivating Contact: A Guide to Building Bridges and Meaningful Connections Between Groups

The United States is in the process of reckoning with many forms of social division, but it is also facing a moment of immense possibility. With deepening divides occurring and being fomented across racial, religious, socioeconomic, partisan, and geographic lines, trust in others has declined and members of distinct groups are more isolated from each other than ever. Many forces seek to exploit these vulnerabilities and stoke fear and anxiety about group differences. Yet our nation's history shows us that, even in the midst of these challenges, Americans from all walks of life have found ways to come together across lines of difference to solve critical community problems.How we choose to respond to group differences is ultimately up to us. We can take steps either to build walls or build bridges in the face of these differences. When we feel insecure, unsafe, or threatened, our initial instinct is to build walls, in an effort to protect ourselves and our groups. This instinctual response can help us to feel more secure and protected in the short term; but one long-term consequence is that we may grow more distrustful and fearful of people who are not like "us" and whom we don't personally know. Worse still, challenging social and economic conditions can exacerbate these tendencies, such that we start to develop competitive narratives that pit "us" against "them" and further deepen existing divisions between groups.Instead, when we build bridges, we take steps to engage with people across lines of difference. Engaging with one another in meaningful and authentic ways often requires us to step outside of our comfort zone, as we begin to share our life stories and experiences openly while attending deeply and respectfully to those shared by others. From interacting with others with this spirit of openness and attentiveness, we invite others into our worlds, just as they invite us into theirs. By doing so, we not only develop greater mutual understanding, but we are also likely to become more invested in each other's lives and to care more about each other's groups—and this emotional investment and caring is what compels us to work toward improving our communities and social institutions to ensure that everyone feels like they belong.In this guide, we describe how to set the stage for people from different backgrounds to engage with each other in ways that foster trust and belonging, while also drawing on their similarities and differences to solve community problems. We review a number of strategies that encourage people from different groups to work together as equals, so that they can share ideas and perspectives, and co-create new initiatives in collaboration and across group divides. We also provide materials that can help organizations begin to envision how they might assess the effectiveness of their contact programs

Language and Intergroup Contact: Investigating the Impact of Bilingual Instruction on Children’s Intergroup Attitudes

This study examined the impact of bilingual versus English-only instruction on the intergroup attitudes of White, English-speaking children in kindergarten through second grade. Replicating prior research, White children generally showed a clear preference toward the ingroup in terms of positive evaluations, friendship preference, and perceived similarity to the self. However, all three effects were significantly smaller among children who were in classrooms with a significant amount of Spanish instruction (i.e. bilingual classes). The smaller preference for the ingroup over the outgroup found in bilingual classes resulted from higher evaluations of, greater selection of friends among, and greater perceived similarity to Latino targets, and not from changes in preference for White ingroup targets. Furthermore, comparisons with English-only classes that had substantial Latino representation shows that the positive impact of bilingual instruction can be only partially explained by the greater representation of Latino children in bilingual classes. Finally, these positive patterns of intergroup attitudes found in bilingual classes were not associated with any negative effects on White children’s personal self-evaluation

Greedy Signal Recovery Review

The two major approaches to sparse recovery are L1-minimization and greedy methods. Recently, Needell and Vershynin developed Regularized Orthogonal Matching Pursuit (ROMP) that has bridged the gap between these two approaches. ROMP is the first stable greedy algorithm providing uniform guarantees. Even more recently, Needell and Tropp developed the stable greedy algorithm Compressive Sampling Matching Pursuit (CoSaMP). CoSaMP provides uniform guarantees and improves upon the stability bounds and RIC requirements of ROMP. CoSaMP offers rigorous bounds on computational cost and storage. In many cases, the running time is just O(NlogN), where N is the ambient dimension of the signal. This review summarizes these major advances

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds $$m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$$ where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples

Atoms of all channels, unite! Average case analysis of multi-channel sparse recovery using greedy algorithms

This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that have the largest $p$-correlation while the second one, p-simultaneous matching pursuit (p-SOMP), is a generalisation of an algorithm studied by Tropp. We first provide exact recovery conditions as well as worst case analyses of all algorithms. The results, expressed using the standard cumulative coherence, are very reminiscent of the single channel case and, in particular, impose stringent restrictions on the dictionary. We unlock the situation by performing an average case analysis of both algorithms. First, we set up a general probabilistic signal model in which the coefficients of the atoms are drawn at random from the standard gaussian distribution. Second, we show that under this model, and with mild conditions on the coherence, the probability that p-thresholding and p-SOMP fail to recover the correct components is overwhelmingly small and gets smaller as the number of channels increases. Furthermore, we analyse the influence of selecting the set of correct atoms at random. We show that, if the dictionary satisfies a uniform uncertainty principle, the probability that simultaneous OMP fails to recover any sufficiently sparse set of atoms gets increasingly smaller as the number of channels increases

Changing the ideological roots of prejudice: Longitudinal effects of ethnic intergroup contact on social dominance orientation

Social Dominance Orientation (SDO) has been reported to be strongly related to a multitude of intergroup phenomena, but little is known about situational experiences that may influence SDO. Drawing from research on intergroup contact theory, we argue that positive intergroup contact is able to reduce SDO-levels. The results of an intergroup contact intervention study among high school students (Study 1, N=71) demonstrated that SDO-levels were indeed attenuated after the intervention. Furthermore, this intervention effect on SDO was especially pronounced among students reporting a higher quality of contact. A cross-lagged longitudinal survey among adults (Study 2, N=363) extended these findings by demonstrating that positive intergroup contact is able to decrease SDO over time. Moreover, we did not obtain evidence for the idea that people high in SDO would engage less in intergroup contact. These findings indicate that intergroup contact erodes one of the important socio-ideological bases of generalized prejudice and discrimination

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's inequality has been moved to a separate note; other martingale bounds are described in Caltech ACM Report 2011-0