Determinantal Subset Diversity

Worked Example

Let's consider a three-item L-ensemble with items A, B, and C, where the L-matrix is defined as
L = a * [[1, rho, 0], [rho, 1, 0], [0, 0, 1]].
We set the intensity a = 1 and similarity rho = 0.5.

The L matrix evaluates to:

L = [[1.0, 0.5, 0.0], [0.5, 1.0, 0.0], [0.0, 0.0, 1.0]]

The normalizer for the probability distribution is the determinant of I + L, or analytically evaluated as:

1. det(I + L) = ((1 + a)^2 - a^2 * rho^2) * (1 + a)
2. For a = 1, rho = 0.5: Normalizer = ((2)^2 - 1^2 * 0.5^2) * 2 = (4 - 0.25) * 2 = 7.5

Let's compute the probability of subset AB being selected:

1. The submatrix L_AB is formed by taking rows and columns corresponding to items A and B.
2. L_AB = [[1.0, 0.5], [0.5, 1.0]]
3. det(L_AB) = 1.0 * 1.0 - 0.5 * 0.5 = 0.75
4. P(AB) = det(L_AB) / Normalizer = 0.75 / 7.5 = 0.1

This confirms our calculation: the probability of selecting exactly subset AB is 0.1, the expected set size is approximately 1.4333, and the inclusion covariance between A and B is negative (-0.0177), demonstrating repulsion.

Experiment

You can reproduce this behavior in the module interactively.

1. Set Intensity to 1.
2. Set Similarity to 0.5.
3. Set Subset to AB.
4. Observe that the selected set probability is 0.1 and the expected size is 1.433.

Limitations

The chosen L-matrix assumes specific structural relations where items A and B have adjustable similarity, while item C is always completely independent (similarity 0) from both. A more general DPP would allow arbitrary Gram matrices. Also note that if we set similarity to 1, items A and B are perfectly similar and compete so strongly that the probability of selecting both becomes exactly 0.

References

For more details on determinantal point processes, see:
Kulesza, A., & Taskar, B. (2012). Determinantal Point Processes for Machine Learning.