Notes on Graph Cuts with Submodular Edge Weights

Generalizing the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of (|V |1/3) on the approximation factor for the (s, t) cut version of the problem. On the positive side, we propose and compare three approximation algorithms with an overall approximation factor of O(min|V |,p|E| log |V |) that appear to do well in practice

Scalable Semidefinite Programming using Convex Perturbations

Several important machine learning problems can be modeled and solved via semidefinite programs. Often, researchers invoke off-the-shelf software for the associated optimization, which can be inappropriate for many applications due to computational and storage requirements. In this paper, we introduce the use of convex perturbations for semidefinite programs (SDPs). Using a particular perturbation function, we arrive at an algorithm for SDPs that has several advantages over existing techniques: a) it is simple, requiring only a few lines of MATLAB, b) it is a first-order method which makes it scalable, c) it can easily exploit the structure of a particular SDP to gain efficiency (e.g., when the constraint matrices are low-rank). We demonstrate on several machine learning applications that the proposed algorithm is effective in finding fast approximations to large-scale SDPs

Beyond a Dichotomy of Perspectives: Understanding Religion on the Basis of Paul Natorp’s Logic of Boundary

Based on Paul Natorp’s (1854–1924) late post-Neo-Kantian “Logic of Boundary” (German: “Grenzlogik”) I will offer a methodically controlled, non-reductionist and equally anti-essentialist reconstruction of the notion of religion. The pre-eminent objective of this reconstructive work is to overcome the well-known epistemological as well as methodological problem of a dichotomy between inside and outside perspectives on the subject of religion. Differently put, the objective consists in an attempt to demonstrate that there actually is “reason in religion” that is intellectually accessible for academic knowledge production. Following Natorp’s splendid formulation I will argue that religion operates neither ‘within’ nor ‘beyond’ the ‘boundary of humanity’ but exactly on [or ‘in’] this boundary. More precisely, I will explicate that religious praxis (including its specific production of knowledge) from Natorp’s standpoint can be understood as the performative realization, and habitual embodiment of the (contextually concrete) boundary of humanity or human reason itself. Due to its principial self-referentiality this boundary carries the crucial sense of a first and last positive and, therefore, both in theoretical terms definitive and in practical terms eminently instructive notion of boundary with no outside. This paradoxically all-enclosing, positive boundary, while explicitly including life’s inevitable negativity but, nonetheless, able to ideally sublate it, is the reason why the practice of religion, as empirical evidence unmistakably documents, can provide an incommensurably fulfilling, significant and meaningful closure with regards to the innermost self-perception of its practitioners (concerning their self-determination or agency)

A Reaction-Diffusion Model to Capture Disparity Selectivity in Primary Visual Cortex

Decades of experimental studies are available on disparity selective cells in visual cortex of macaque and cat. Recently, local disparity map for iso-orientation sites for near-vertical edge preference is reported in area 18 of cat visual cortex. No experiment is yet reported on complete disparity map in V1. Disparity map for layer IV in V1 can provide insight into how disparity selective complex cell receptive field is organized from simple cell subunits. Though substantial amounts of experimental data on disparity selective cells is available, no model on receptive field development of such cells or disparity map development exists in literature. We model disparity selectivity in layer IV of cat V1 using a reaction-diffusion two-eye paradigm. In this model, the wiring between LGN and cortical layer IV is determined by resource an LGN cell has for supporting connections to cortical cells and competition for target space in layer IV. While competing for target space, the same type of LGN cells, irrespective of whether it belongs to left-eye-specific or right-eye-specific LGN layer, cooperate with each other while trying to push off the other type. Our model captures realistic 2D disparity selective simple cell receptive fields, their response properties and disparity map along with orientation and ocular dominance maps. There is lack of correlation between ocular dominance and disparity selectivity at the cell population level. At the map level, disparity selectivity topography is not random but weakly clustered for similar preferred disparities. This is similar to the experimental result reported for macaque. The details of weakly clustered disparity selectivity map in V1 indicate two types of complex cell receptive field organization

Cooperative Cuts

Combinatorial problems with submodular cost functions have recently drawn interest. In a standard combinatorial problem, the sum-of-weights cost is replaced by a submodular set function. The result is a powerful model that is though very hard. In this talk, I will introduce cooperative cuts, minimum cuts with submodular edge weights. I will outline methods to approximately solve this problem, and show an application in computer vision. If time permits, the talk will also sketch regret-minimizing online algorithms for submodular-cost combinatorial problems. This is joint work with Jeff Bilmes (University of Washington)