Counting dead directions
A dead direction usually does not come alone. A real singular minimum has several collapsing together, the multiplicity that comes with the order. Detecting that at least one is present is easy. Counting them needs a different read, because the obvious one cannot see the count.
The smallest eigenvalue cannot count
The cheapest reads watch a single number: $\sigma_{\min}$ of the activations, or $\lambda_{\min}(G)$ of the Fisher. By construction that number is the shallowest direction alone. If three directions are dying together, the smallest eigenvalue still reports only the shallowest of the three, and its decay slope is $2$ whether one direction is dead or ten. It is rank-blind.
Volume counts them
To count, add them up, and the right adder is a volume. The determinant of $G$ multiplies every eigenvalue together, one factor per direction, so it behaves like the volume of a box with one side per direction: shrink any side and the whole volume shrinks with it, no matter which side it was. Taking the log turns that product into a sum, written $\log\det^{+} G$ (the ${}^{+}$ says exact zeros, the gauge directions that never had curvature to lose, are left out of the product). Now no dying direction can hide behind the others: with $r$ dead directions each falling as $t^2$, the volume falls as $t^{2r}$, a slope of $2r$, which is $r$ times the single-direction slope. Divide one by the other and the count drops out, read from a single determinant.
Turn the rank-deficit $r$. The smallest-eigenvalue slope sits at $2$ and never moves: it cannot tell $r$ dead directions from one. The volume slope is $2r$, and dividing by the single-direction slope recovers $r$ exactly. The smallest eigenvalue says "something is dead here"; the volume says "how many."
What the count is, exactly
The number the volume recovers is the rank-deficit $r$: how many directions are collapsing together at this layer. On controlled networks with a known rank-deficit, the volume slope tracks it across $r = 1, 2, 3, 4$, where a smallest-eigenvalue monitor stays flat throughout.1 The measured runs are below: forty cells per rank, and the slope ratio lands on $r$ each time.
It is tempting to equate that count with the multiplicity $m$ from earlier, and on clean cells the two agree: when all $r$ directions die at the same order on one shared piece of the singular set, the rank-deficit is the analytic multiplicity. On a real network the accurate name is the dead-subspace dimension, the count of directions that have genuinely collapsed, with two kinds of impostor excluded: gauge directions, flat because of a symmetry of the architecture, and near-zero eigenvalues that are numerical floor rather than structure. That floor-aware count equals $m$ only under the shared-locus assumption, and keeping the two ideas separate is what lets the count stay meaningful on networks where structures of different kinds mix.2 A single eigenvalue tells you the order; the whole spectrum, through its volume, tells you how many directions share it; and turning per-direction counts into the model's one global complexity is the assembly problem of adding up the orders#23.
Still open / underspecified. The clean $2r$ slope assumes the $r$ dead directions approach at the same order and that the spectrum is resolved well enough to separate the shrinking eigenvalues from the floor. Mixed kinds and orders are no longer the obstacle they once were, the typed assembly of adding up the orders#23 sorts directions before counting them, but two edges remain: finite-sample noise can lift small eigenvalues off the floor into an effective count that needs care, and a multiplicity split across several equally flat pieces of the singular set still under-counts.
- The rank-multiplicative volume identity, and its validation across rank-deficits, are from Tejas Pradeep Shirodkar and P. J. Narayanan, Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity, arXiv:2606.21158 (2026). ↩︎
- The dead-subspace dimension, its floor-aware definition, and the conditions under which it equals the analytic multiplicity are from Tejas Pradeep Shirodkar, Measuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment, arXiv:2607.00603 (2026). ↩︎