I Dropped a Neural Net

Hyunwoo Park

cs.LGFeb 23, 2026

I Dropped a Neural Net

Authors: Hyunwoo Park

Organizations: Carnegie Mellon University

Abstract

A recent Dwarkesh Patel podcast with John Collison and Elon Musk featured an interesting puzzle from Jane Street: they trained a neural net, shuffled all 96 layers, and asked to put them back in order. Given unlabelled layers of a Residual Network and its training dataset, we recover the exact ordering of the layers. The problem decomposes into pairing each block's input and output projections ( $48!$ possibilities) and ordering the reassembled blocks ( $48!$ possibilities), for a combined search space of $(48!)^2 \approx 10^{122}$ , which is more than the atoms in the observable universe. We show that stability conditions during training like dynamic isometry leave the product $W_{\text{out}} W_{\text{in}}$ for correctly paired layers with a negative diagonal structure, allowing us to use diagonal dominance ratio as a signal for pairing. For ordering, we seed-initialize with a rough proxy such as delta-norm or $\|W_{\text{out}}\|_F$ then hill-climb to zero mean squared error.

Figures & tables

Figure 1: Architecture. The 48-block ResNet. Each block applies input layer, ReLU, output layer, then a residual connection. The final layer produces the scalar output $\hat{y}$ .

Panel of Figure 2 — Figure 2: Dataset overview. Features and outputs are normalized.

Panel of Figure 3 — Figure 3: Pairing via diagonal dominance. The diagonal dominance ratio achieves complete separation between correct pairs ( $d\in[1.76,3.28]$ ) and incorrect pairs ( $d\in[0.00,0.58]$ ), with a gap of $1.18$ .

Figure 4: Correct vs. incorrect pairings. Top: Correctly paired matrices show negative diagonal structure. Bottom: Incorrect pairings with no structure.

Figure 5: All 48 product matrices from the original model, ordered by delta-norm $\delta_{k}$ (Section 4.1 ). Every block exhibits a negative diagonal structure. Earlier blocks (top-left) and later blocks (bottom-right). Traces range from $-13.5$ to $-7.4$ . Compare with Figure 7 .

Round	Swaps	MSE	Cum. swaps
0 (BT init)	–	0.002989	0
1	21	0.001025	21
2	7	0.000553	28
3	5	0.000200	33
4	3	0.000052	36
5	1	0.000000	37

Table 1: Bubble repair convergence from BT ordering. Converges to MSE

=0

in 5 rounds with 37 swaps.

Panel of Figure 6 — Figure 6: Ordering pipeline progression. Bar height = ground-truth position (1–48)

Round	Swaps	MSE	Cum. swaps
0 (seed)	–	0.036794	0
1	23	0.021690	23
2	20	0.012919	43
3	21	0.008559	64
4	15	0.005873	79
5	11	0.004097	90

Table 2: Hill-climb from delta-norm seed (no BT). Converges to exact solution in 13 rounds / 122 swaps.

Round	Swaps	MSE	Cum. swaps
0 (seed)	–	0.075851	0
1	30	0.005936	30
2	16	0.003397	46
3	12	0.001405	58
4	9	0.000568	67
5	4	0.000223	71

Table 3: Hill-climb from

\|W_{\text{out}}\|_{F}

seed (data-free, no BT). Starting from MSE

=0.075851

, converges in 6 rounds / 72 swaps. Despite higher initial MSE, convergence is

2\times

faster in rounds and

1.7\times

fewer swaps, and is data-free .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Product matrices from a freshly trained model. All 48 matrices $W_{\mathrm{out}}^{(k)}W_{\mathrm{in}}^{(k)}$ , sorted by Frobenius norm.

Figure 8: ReLU activation fraction across blocks. Per-block ReLU activation probability

Figure 9: Block properties by recovered position. Top-left: singular value spectra of $W_{\mathrm{out}}W_{\mathrm{in}}$ , colored by position (purple = early, yellow = late). Top-right: effective rank vs. position. Bottom-left: delta-norm increases with depth. Bottom-right: ReLU active fraction

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Rank Collapse, Fixed Points, and the Renormalization Group Structure of MLP Residual Networks

Parviz Haggi-Mani, Irina Rish

Universit´e de Montr´eal · Mila - Quebec AI Institute

Multilayer Perceptrons Neural Network Training Dynamics

Open paper AskPDF arXiv

The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.

Abstract

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