Determinantal Point Process.

Review of 3 papers from Kulesza and Taskar on DPP.

Determinantal Point ProcessP(A \subseteq Y)=det(K_{A}), where Y is a discrete set of points, and K is a kernel on Y.

The benefit of DPP is that its partition function can be computed in closed form, as well as marginals and conditionals. To sample from a DPP, we need to use the eigen-decomposition of K. It captures repulsion between points to enforce diversity.

The 3 papers of Taskar looks at a few variations on this theme:

  1. Structured DPP. K can be decomposed as K_{ij}=q(y_{i})q(y_{j})\phi(y_{i})^{t}\phi(y_{j}), where we think of q as the (scalar) quality of a point and \phi as a feature vector. The structure comes in where we can have y=\{y_i\} and decompose q(y)=\Pi q(y_i) and \phi(y)=\sum \phi(y_i). Applied it to estimate pose (odd choice). Not sure the details.
  2. k-DPP. The regular DPP can generate subsets of points of various sizes. k-DPP basically constrains it to generate exactly k points. Applied to find sets of diverse images. The learning is over a convex sum of different kernels (i.e. learn the weights on a training set).
  3. Learning. Treat the quality q(y)=exp(\theta^{t}f(y)), and optimize over \theta from training set. Applied to text summarization.
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