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We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry, these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity trade-off curve in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume that the underlying matrices have independent and identically distributed Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are universal across a range of underlying matrix ensembles. We ran millions of linear programs using random matrices spanning several matrix ensembles and problem sizes; visually, the empirical phase transitions do not depend on the ensemble, and they agree extremely well with the asymptotic theory assuming Gaussianity. Careful statistical analysis reveals discrepancies that can be explained as transient terms, decaying with problem size. The experimental results are thus consistent with an asymptotic large-n universality across matrix ensembles; finite-sample universality can be rejected.

Matrices of low rank can be uniquely determined from fewer linear measurements, or entries, than the total number of entries in the matrix. Moreover, there is a growing literature of computationally efficient algorithms which can recover a low rank matrix from such limited information; this process is typically referred to as matrix completion. We introduce a particularly simple yet highly efficient alternating projection algorithm which uses an adaptive stepsize calculated to be exact for a restricted subspace. This method is proven to have near-optimal order recovery guarantees from dense measurement masks and is observed to have average case performance superior in some respects to other matrix completion algorithms for both dense measurement masks and entry measurements. In particular, this proposed algorithm is able to recover matrices from extremely close to the minimum number of measurements necessary. [PUBLICATION ABSTRACT]

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.

Consider an underdetermined system of linear equations y = Ax with known y and d x n matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y = Ax. In general, this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. We explain this by the theory of convex polytopes. Let ajdenote the jth column of A, 1 ≤ j ≤ n, let a0=0 and P denote the convex hull of the aj. We say the polytope P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. We also consider weak neighborliness, where the overwhelming majority of k-sets of ajs not containing 0 span a face of P. This implies that most nonnegative vectors x with k nonzeros are uniquely recoverable from y = Ax by linear programming. Numerous corollaries follow by invoking neighborliness results. For example, for most large n by 2n underdetermined systems having a solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.

Objective:Higher cardiovascular burden and peripheral inflammation are associated with small vessel vascular disease, a predominantly dysexecutive cognitive profile, and a higher likelihood of conversion to vascular dementia. The digital clock drawing test, a digitized version of a standard neuropsychological tool, is useful in identifying cognitive dysfunction related to vascular etiology. However, little is known about the specific cognitive implications of vascular risk, peripheral inflammation, and varying levels of overall brain integrity. The current study aimed to examine the role of cardiovascular burden, peripheral inflammation, and brain integrity on digitally acquired clock drawing latency and graphomotor metrics in non-demented older adults.Participants and Methods:The final prospectively recruited IRB-consented participant sample included 184 non-demented older adults (age: 69±6 years, education: 16±3 years, 46% female, 94% white) who completed digital clock drawing, vascular assessment, blood draw, and brain MRI. Digital clock drawing variables of interest included: total completion time (TCT), pre-first hand latency (PFHL), digit misplacement, hour hand distance from center, and clock face area (CFA). Cardiovascular burden was calculated using the revised version of the Framingham Stroke Risk Profile (FSRP-10). Peripheral inflammation was operationalized using interleukin (IL)-6, IL-8, IL-10, tumor necrosis factor alpha (TNF-a), and high sensitivity C-reactive protein (hsCRP). The brain integrity composite was comprised of bilateral entorhinal cortex volume, bilateral ventricular volume, and whole brain leukoaraiosis.Results:Over and above age and cognitive reserve, hierarchical regressions showed FSRP-10, inflammatory markers, and brain integrity explained an additional 13.3% of the variance in command TCT (p< 0.001), with FSRP-10 (p=0.001), IL-10 (p= 0.019), and hsCRP (p= 0.019) as the main predictors in the model. FSRP-10, inflammatory markers, and brain integrity explained an additional 11.7% of the variance in command digit misplacement (p= 0.009), with findings largely driven by FSRP-10 (p< 0.001).Conclusions:Overall, in non-demented older adults, subtle behavioral nuances seen in digital clock drawing metrics (i.e., total completion time and digit misplacement) are partly explained by cardiovascular burden, peripheral inflammation, and brain integrity over and above age and cognitive reserve. These nuanced behaviors on digitally acquired clock drawing may associate with an emergent disease process or overall vulnerability.Funding sources: Barber Fellowship; K07AG066813; R01 AG055337; R01 NR014810; American Psychological Foundation Dissertation Award; APA Dissertation Research Award

Objectives Parkinson's disease (PD) is a multisystem movement disorder associated with sleep disturbance and depression. Sleep disturbances and depression severity share a bidirectional association. This association may be greater in individuals who are more vulnerable to the deleterious consequences of sleep disturbance and depression severity. We investigated whether the association between sleep disturbances and depression severity is greater in patients with PD than in matched controls (MC). Materials and Methods The study sample (N = 98) included 50 patients with idiopathic PD and 48 age‐, race‐, sex‐, and education‐matched controls. Sleep disturbances were assessed using self‐reported total sleep time (TST) on the Pittsburgh Sleep Quality Index, the sleep item on the Beck Depression Inventory, 2nd ed. (BDI‐II), and the Insomnia Severity Index total score. Depression severity was assessed using the BDI‐II total score, excluding the sleep item. Spearman's correlations, chi‐squared tests, and multiple regression were used to assess associations between sleep disturbances and depression severity in PD and MC. Fisher's Z transformation was used to test whether the association between sleep disturbances and depression severity was stronger in patients with PD. Results Shorter TST, sleeping less than usual, and insomnia severity were associated with depression severity in the total sample, rs(94) = −0.35, p = .001; rs(71) = 0.51, p < .001; rs(78) = −0.47, p < .001; rs(98) = 0.46, p < .001, respectively. The association between shorter TST and depression severity was greater in patients with PD than it was in MC, p < .05. Conclusion Short TST may be an important marker, predictor, or consequence of depression severity in patients with Parkinson's disease. Self‐reported short total sleep time was more strongly associated with depression in patients with Parkinson's disease (PD) than matched controls. Short sleep may be an important marker, predictor, or consequence of depression severity in patients with PD.

Let A be a d x n matrix and T=Tn-1be the standard simplex in Rn. Suppose that d and n are both large and comparable: d ≈ δn, δ ∈ (0, 1). We count the faces of the projected simplex AT when the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn. We derive$\\rho _{N}(\\delta)>0$with the property that, for any$\\rho <\\rho _{N}(\\delta)$, with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0 ≤ k ≤ pd. This implies that P is$\\lfloor \\rho d\\rfloor \\text{-}\\text{neighborly}$, and its skeleton$Skel_{\\lfloor \\rho d\\rfloor}(P)$is combinatorially equivalent to$Skel_{\\lfloor \\rho d\\rfloor}(T)$. We also study a weaker notion of neighborliness where the numbers of k-dimensional faces fk(P)≥ fk(T)(1-ε). Vershik and Sporyshev previously showed existence of a threshold$\\rho _{VS}(\\delta)>0$at which phase transition occurs in k/d. We compute and display ρVSand compare with ρN. Corollaries are as follows. (1) The convex hull of n Gaussian samples in Rd, with n large and proportional to d, has the same k-skeleton as the (n - 1) simplex, for$k<\\rho _{N}(d/n)d(1+o_{P}(1))$. (2) There is a \"phase transition\" in the ability of linear programming to find the sparsest nonnegative solution to systems of underdetermined linear equations. For most systems having a solution with fewer than ρVS(d/n)d(1+o(1)) nonzeros, linear programming will find that solution.

Chronic musculoskeletal pain including knee osteoarthritis (OA) is a leading cause of disability worldwide. Previous research indicates ethnic-race groups differ in the pain and functional limitations experienced with knee OA. However, when socioenvironmental factors are included in analyses, group differences in pain and function wane. Pain-related brain structures are another area where ethnic-race group differences have been observed. Environmental and sociocultural factors e.g., income, education, experiences of discrimination, and social support influence brain structures. We investigate if environmental and sociocultural factors reduce previously observed ethnic-race group differences in pain-related brain structures. Data were analyzed from 147 self-identified non-Hispanic black (NHB) and non-Hispanic white (NHW), middle and older aged adults with knee pain in the past month. Information collected included health and pain history, environmental and sociocultural resources, and brain imaging. The NHB adults were younger and reported lower income and education compared to their NHW peers. In hierarchical multiple regression models, sociocultural and environmental factors explained 6–37% of the variance in pain-related brain regions. Self-identified ethnicity-race provided an additional 4–13% of explanatory value in the amygdala, hippocampus, insula, bilateral primary somatosensory cortex, and thalamus. In the rostral/caudal anterior cingulate and dorsolateral prefrontal cortex, self-identified ethnicity-race was not a predictor after accounting for environmental, sociocultural, and demographic factors. Findings help to disentangle and identify some of the factors contributing to ethnic-race group disparities in pain-related brain structures. Numerous arrays of environmental and sociocultural factors remain to be investigated. Further, the differing sociodemographic representation of our NHB and NHW participants highlights the role for intersectional considerations in future research.

Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004 there have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS—exact reconstruction from seemingly undersampled measurements—it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry.