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"Neighborliness."
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Annie and Snowball and the thankful friends
Annie wants to have a big Thanksgiving dinner and thinks about who she can invite to share it with her.
Book
UNIVERSALITY IN POLYTOPE PHASE TRANSITIONS AND MESSAGE PASSING ALGORITHMS
We consider a class of nonlinear mappings FA,N in ℝN indexed by symmetric random matrices A ∈ ℝN×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333–366]. Within information theory, they are known as \"approximate message passing\" algorithms. We study the high-dimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal; that is, it depends only on the first two moments of the entries of A, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.
Journal Article
What's new, Daniel?
A curious little boy explores his neighborhood, finding out what is new with his friends and neighbors.
Book
The Oneness of Love in IWorks of Love/I
Kierkegaard’s claim that God just is love implies that love is ultimately one reality. Indeed, on more than one occasion, Kierkegaard will make this point explicitly as well as implicitly by frequently asserting the oneness of love. For example, early on in Works of Love, he states plainly that “this love for the neighbor is not related as a type to other types of love. Erotic love is defined by the object; friendship is defined by the object; only love for the neighbor is defined by love”. What Kierkegaard means by this is that preferential loves are defined by a factor in addition to love itself: the object of that love. Neighbor-love is defined by love itself, which takes as its object the neighbor, or in other words, “unconditionally every human being”. Preferential loves are specified as it were by the person loved in this manner. Neighbor-love is not related as a type to other types of love in that neighbor-love is paradigmatic love; preferential loves are specified, but as recent commentators have shown, are not thereby precluded from also being or filtered by or infused by or coincident with neighbor-love as well. The point of this passage is that there are not distinct, enumerated types of love that, taken together, can be amalgamated into something called “love”, which would be inclusive of distinct kinds. The current paper argues that neighbor-love is meant to be thought of as paradigmatic. Therefore, as a paradigmatic unity, it will also exhibit qualities ordinarily associated with preferential love. Put differently, my claim is that we have reason to conclude that, in the end, features of preferential love will be manifest in neighbor-love just as surely as neighbor-love has an effect on preferential love. I wish to take seriously the claim of Works of Love that, ultimately, love is one. Love, being one, is not comprised of distinct types or subsets. I demonstrate the importance of this point by explaining how all love has its ultimate origin in God (and God just is love). While seemingly a truism, I argue from a variety of passages that the oneness of love has multiple implications throughout the text, implications that further support the theory that neighbor-love is not an alternative to, but rather encompasses features of, preferential loves.
Journal Article
May I come in?
Raccoon does not want to be alone on a stormy night but his neighbors, Possum, Quail, and Woodchuck, each tell him they have no room to spare.
Book
Counting faces of randomly projected polytopes when the projection radically lowers dimension
Let Q=QNQ = Q_N denote either the NN-dimensional cross-polytope CNC^N or the N−1N-1-dimensional simplex TN−1T^{N-1}. Let A=An,NA = A_{n,N} denote a random orthogonal projector A:RN↦bRnA: \\mathbf {R}^{N} \\mapsto bR^n. We compare the number of faces fk(AQ)f_k(AQ) of the projected polytope AQAQ to the number of faces of fk(Q)f_k(Q) of the original polytope QQ. We concentrate on the case where nn and NN are both large, but nn is much smaller than NN; in this case the projection dramatically lowers dimension. We consider sequences of triples (k,n,N)(k,n,N) where N=NnN = N_n is not exponentially larger than nn. We identify thresholds of the form const⋅nlog(n/N)const \\cdot n \\log (n/N) where the relationship of fk(AQ)f_k(AQ) and fk(Q)f_k(Q) changes abruptly. These properties of polytopes have significant implications for neighborliness of convex hulls of Gaussian point clouds, for efficient sparse solution of underdetermined linear systems, for efficient decoding of random error correcting codes and for determining the allowable rate of undersampling in the theory of compressed sensing. The thresholds are characterized precisely using tools from polytope theory, convex integral geometry, and large deviations. Asymptotics developed for these thresholds yield the following, for fixed ϵ>0\\epsilon > 0. With probability tending to 1 as nn, NN tend to infinity: (1a) for k>(1−ϵ)⋅n[2eln(N/n)]−1k > (1-\\epsilon ) \\cdot n [2e\\ln (N/n)]^{-1} we have fk(AQ)=fk(Q)f_k(AQ) = f_k(Q), (1b) for k>(1+ϵ)⋅n[2eln(N/n)]−1k > (1 +\\epsilon ) \\cdot n [2e\\ln (N/n)]^{-1} we have fk(AQ)>fk(Q)f_k(AQ) > f_k(Q), with E{\\mathcal E} denoting expectation, (2a) for k>(1−ϵ)⋅n[2ln(N/n)]−1k > (1-\\epsilon ) \\cdot n [2\\ln (N/n)]^{-1} we have Efk(AQ)>(1−ϵ)fk(Q){\\mathcal E} f_k(AQ) > (1-\\epsilon ) f_k(Q), (2b) for k>(1+ϵ)⋅n[2ln(N/n)]−1k > (1 +\\epsilon ) \\cdot n [2\\ln (N/n)]^{-1} we have Efk(AQ)>ϵfk(Q){\\mathcal E} f_k(AQ) > \\epsilon f_k(Q). These asymptotically sharp transitions in the behavior of face numbers as kk varies relative to nlog(N/n)n \\log (N/n) are proven, interpreted, and related to the above-mentioned applications.
Journal Article
Look up!
When a girl in a wheelchair calls to people far below to look up and see her, one finds a way to brighten her day.
Book
Candy shop
When a boy and his aunt find that a bigot has written hurtful words on the sidewalk just outside the candy shop owned by \"Miz Chu\", a new immigrant from Taiwan, they set out to comfort her.
Book
