k-DPPs Lose More Identifiability Than Full DPPs Do

When a diversity model is forced to pick exactly k items, its hidden parameters become harder to pin down. This paper studies determinantal point processes through a spectral decomposition, where eigenvalues control how many items appear and the eigenvector directions control which items appear together. For the fixed-size version, called a k-DPP, the authors show that the spectral part is identifiable only up to a common scale, and the orientation part is recoverable only through squared minors of the eigenvector matrix. They describe the identifiability gap with three explicit symmetries: scale, sign similarity, and eigenspace rotation. They also prove a dimension-counting theorem: whenever binom(N,k) is smaller than N(N+1)/2, additional continuous non-identifiability must exist. In the full DPP, by contrast, the only non-identifiability comes from a discrete sign similarity.

A DPP has two hidden dials. One sets how many items enter the set, and the other sets which items like to appear together. That split makes the model feel orderly, almost simple. But the fixed-size version, the k-DPP, strips away more information than a casual glance suggests. Once you force the model to choose exactly k items, the overall scale of the spectrum no longer matters, and the eigenvector directions no longer show up in full. What remains is thinner and stranger: only certain squared pieces of the orientation survive. If you have ever trusted a model because it looked compact, this is the warning sign. Compact can also mean harder to read.

Two hidden knobs, one of them gone

Writing the kernel as L = UΛU^T splits the model into two parts. The eigenvalues in Λ feed the size distribution through elementary symmetric polynomials, a family of sums that turns the spectrum into chances for each possible set size. The matrix U carries the eigenspace orientation, and that part decides what happens once you lock the set size at k. In the full DPP, that split leaves only a discrete sign ambiguity: flip eigenvector signs and nothing changes. In a k-DPP, the picture changes more sharply. The spectrum becomes identifiable only up to a common scale, while the orientation survives only through squared minors of the eigenvector matrix. So the fixed-size model does not merely lose a little detail; it changes which details can be recovered at all.

How the same set can hide more than one kernel

The key move is conditioning on cardinality, because that removes the part of the model that tells you how likely each size is. Once every sample must have exactly k elements, multiplying all eigenvalues by the same constant leaves the k-DPP unchanged, so the spectrum can no longer fix its absolute scale. The orientation part blurs for a different reason. The model keeps the squared minors of U, not the raw signs or every possible rotation, so several different-looking eigenvector layouts can collapse to the same fixed-size law. That is why the analysis separates scale, sign similarity, and eigenspace rotation instead of treating them as one vague ambiguity.

Why fixed-size data can still leave too many answers

This matters because fixed-size data can make a model look more certain than it really is. If you only observe k-sized sets, you cannot expect to recover the full kernel uniquely, since a common scale can vanish and the orientation can survive only in squared form. The dimension-count theorem sharpens that warning: whenever binom(N,k) is smaller than N(N+1)/2, extra continuous non-identifiability must exist. In plain terms, the space of possible fixed-size outputs is too small to pin down all the underlying knobs. For anyone fitting k-DPPs, that means a neat numerical answer may hide several equally valid ones.

Where the boundary sits

The fixed-size model therefore behaves less like a smaller DPP and more like a different creature with its own blind spots. A kernel that looks unique on paper can still sit inside an equivalence class once you condition on size, because the scale drops away and the orientation only speaks through squared minors. The cleanest next test sits on the boundary of the theorem: cases where binom(N,k) just meets or falls below N(N+1)/2. That is where the model stops having enough distinct fixed-size outcomes to pin down all its hidden knobs. At that edge, identifiability turns from a tidy algebra question into a question about what fixed-size data can truly tell you.