From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma 8.3 that appear in the published versio

The achievable performance of convex demixing

Demixing is the problem of identifying multiple structured signals from a superimposed, undersampled, and noisy observation. This work analyzes a general framework, based on convex optimization, for solving demixing problems. When the constituent signals follow a generic incoherence model, this analysis leads to precise recovery guarantees. These results admit an attractive interpretation: each signal possesses an intrinsic degrees-of-freedom parameter, and demixing can succeed if and only if the dimension of the observation exceeds the total degrees of freedom present in the observation

Convexity in source separation: Models, geometry, and algorithms

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems. Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work

Sharp recovery bounds for convex demixing, with applications

Demixing refers to the challenge of identifying two structured signals given only the sum of the two signals and prior information about their structures. Examples include the problem of separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis, and the problem of decomposing an observed matrix into a low-rank matrix plus a sparse matrix. This paper describes and analyzes a framework, based on convex optimization, for solving these demixing problems, and many others. This work introduces a randomized signal model which ensures that the two structures are incoherent, i.e., generically oriented. For an observation from this model, this approach identifies a summary statistic that reflects the complexity of a particular signal. The difficulty of separating two structured, incoherent signals depends only on the total complexity of the two structures. Some applications include (i) demixing two signals that are sparse in mutually incoherent bases; (ii) decoding spread-spectrum transmissions in the presence of impulsive errors; and (iii) removing sparse corruptions from a low-rank matrix. In each case, the theoretical analysis of the convex demixing method closely matches its empirical behavior.Comment: 51 pages, 13 figures, 2 tables. This version accepted to J. Found. Comput. Mat

Scaling and universality in the 2D Ising model with a magnetic field

The scaling function of the 2D Ising model in a magnetic field on the square and triangular lattices is obtained numerically via Baxter's variational corner transfer matrix approach. The use of the Aharony-Fisher non-linear scaling variables allowed us to perform calculations sufficiently away from the critical point to obtain very high precision data, which convincingly confirm all predictions of the scaling and universality hypotheses. The results are in excellent agreement with the field theory calculations of Fonseca and Zamolodchikov as well as with many previously known exact and numerical results for the 2D Ising model. This includes excellent agreement with the classic analytic results for the magnetic susceptibility by Barouch, McCoy, Tracy and Wu, recently enhanced by Orrick, Nickel, Guttmann and Perk.Comment: 5 pages, 1 figur

The synthetic triterpenoid CDDO-methyl ester modulates microglial activities, inhibits TNF production, and provides dopaminergic neuroprotection

<p>Abstract</p> <p>Background</p> <p>Recent animal and human studies implicate chronic activation of microglia in the progressive loss of CNS neurons. The inflammatory mechanisms that have neurotoxic effects and contribute to neurodegeneration need to be elucidated and specifically targeted without interfering with the neuroprotective effects of glial activities. Synthetic triterpenoid analogs of oleanolic acid, such as methyl-2-cyano-3,12-dioxooleana-1,9-dien-28-oate (CDDO-Me, RTA 402) have potent anti-proliferative and differentiating effects on tumor cells, and anti-inflammatory activities on activated macrophages. We hypothesized that CDDO-Me may be able to suppress neurotoxic microglial activities while enhancing those that promote neuronal survival. Therefore, the aims of our study were to identify specific microglial activities modulated by CDDO-Me <it>in vitro</it>, and to determine the extent to which this modulation affords neuroprotection against inflammatory stimuli.</p> <p>Methods</p> <p>We tested the synthetic triterpenoid methyl-2-cyano-3,12-dioxooleana-1,9-dien-28-oate (CDDO-Me, RTA 402) in various <it>in vitro </it>assays using the murine BV2 microglia cell line, mouse primary microglia, or mouse primary peritoneal macrophages to investigate its effects on proliferation, inflammatory gene expression, cytokine secretion, and phagocytosis. The antioxidant and neuroprotective effects of CDDO-Me were also investigated in primary neuron/glia cultures from rat basal forebrain or ventral midbrain.</p> <p>Results</p> <p>We found that at low nanomolar concentrations, treatment of rat primary mesencephalon neuron/glia cultures with CDDO-Me resulted in attenuated LPS-, TNF- or fibrillar amyloid beta 1–42 (Aβ1–42) peptide-induced increases in reactive microglia and inflammatory gene expression without an overall effect on cell viability. In functional assays CDDO-Me blocked death in the dopaminergic neuron-like cell line MN9D induced by conditioned media (CM) of LPS-stimulated BV2 microglia, but did not block cell death induced by addition of TNF to MN9D cells, suggesting that dopaminergic neuroprotection by CDDO-Me involved inhibition of microglial-derived cytokine production and not direct inhibition of TNF-dependent pro-apoptotic pathways. Multiplexed immunoassays of CM from LPS-stimulated microglia confirmed that CDDO-Me-treated BV2 cells produced decreased levels of specific subsets of cytokines, in particular TNF. Lastly, CDDO-Me enhanced phagocytic activity of BV2 cells in a stimulus-specific manner but inhibited generation of reactive oxygen species (ROS) in mixed neuron/glia basal forebrain cultures and dopaminergic cells.</p> <p>Conclusion</p> <p>The neuroimmune modulatory properties of CDDO-Me indicate that this potent antioxidant and anti-inflammatory compound may have therapeutic potential to modify the course of neurodegenerative diseases characterized by chronic neuroinflammation and amyloid deposition. The extent to which synthetic triterpenoids afford therapeutic benefit in animal models of Parkinson's and Alzheimer's disease deserves further investigation.</p

The Achievable Performance of Convex Demixing

Demixing is the problem of identifying multiple structured signals from a superimposed, undersampled, and noisy observation. This work analyzes a general framework, based on convex optimization, for solving demixing problems. When the constituent signals follow a generic incoherence model, this analysis leads to precise recovery guarantees. These results admit an attractive interpretation: each signal possesses an intrinsic degrees-of-freedom parameter, and demixing can succeed if and only if the dimension of the observation exceeds the total degrees of freedom present in the observation