NOTE-653: Revisiting Model Stitching to Compare Neural Representations

Contribution

Lenc and Vedaldi (LIT-363) introduced stitching layers to test whether two networks’ features are equivalent up to a learned linear map. This paper turns that test into a general-purpose measurement. It defines a task-relative, asymmetric stitching penalty and argues that it shows what CKA cannot: two representations can be interchangeable for the task while CKA calls them far apart, and one representation can be better than another, which a symmetric similarity cannot say. It then reports three empirical regularities: networks trained from different seeds stitch at every layer (stitching connectivity); different training methods yield early layers that stitch into each other; and representations trained with more data, and near the top with more width or training time, improve a weaker network when plugged into it.

Key insight

Ask not how similar two representations are, but whether one can stand in for the other: replace the bottom of a trained network by the candidate representation, allow only a cheap trained adapter between them, and measure the change in the network’s own loss. The answer is in the task’s units, and it can be directional: r may replace A’s layers when A’s layers cannot replace r.

Assumptions

• The stitching family is “simple”. S is a 1×1 convolution with a BatchNorm before and after it (§2, App. A.4), or a 768×768 token-wise linear map for the ViT. Simplicity is argued, not defined. The control is a randomly initialised bottom network, whose penalty rises to about 34 points (CIFAR-10) and about 45 (ImageNet) at the deepest layers, but is near zero for the first fifth of the layers (Fig. 2A–B, Fig. 5).
• The infimum is a training run. L_ℓ(r; A) = inf over s ∈ S of L(A_{>ℓ} ∘ s ∘ r) (eq. 1) is approximated by fitting s with Adam, cosine decay, learning rate 0.001, on the task’s training set; the penalty is measured on its test set. So the measured penalty mixes approximation, optimisation and generalisation.
• The top network is fixed. Only s is trained. A_{>ℓ} is A’s own upper network, unchanged.
• One task. The loss is the test error of the task A was trained for (CIFAR-10 or ImageNet classification). The authors say so: “model stitching depends on the downstream task” (§3).
• Mostly same architecture. r = B_{≤ℓ} for a B of A’s architecture, except in the width experiment, where the channel counts differ. Stitching is between ResNet blocks only.
• Setting. CIFAR-10 with ResNet-18 (64 first-layer filters) unless stated; ImageNet with ResNet-50, using published PyTorch, VISSL and DINO checkpoints; a ViT trained on CIFAR-5m and stitched on CIFAR-10. The CIFAR networks are trained with random crops and horizontal flips.

Key results

All values are read off plots; none is tabulated in the paper.

• Definition of the penalty (§2). Penalty = L_ℓ(r; A) − L(A). The text then says: “If the penalty is non-negative, then we say that the representation r is at least as good as the first ℓ layers of A.” With a loss, a non-negative penalty means r does no better; the intended condition must be non-positive. The slip is in both arXiv v1 and the camera-ready. The figures use the right sign: “better representation improves performance” is plotted as a negative penalty (Fig. 2C).
• Table 1 (qualitative). CKA: “varies (can be 0)” for different initialisations, “0.35–0.9” for self-supervision, “far (can be 0 for data, 0.7 for width)”. Stitching: “close (up to 3% error)”, “close (up to 5% error)”, “better”.
• Stitching connectivity (Fig. 2A, §4). For ViT, ResNet-18 at 0.5×, 1× and 2× width, ResNet-164, Myrtle-CNN, and ResNet-18 on disjoint training sets, the penalty stays between about −1 and +5 points at every layer; the ViT is the largest, about 5 at mid depth. One pair of seeds per architecture. Definition 1 (“low penalty”, “test loss comparable to A”) and Conjecture 2 (“for natural architectures and data-distributions … stitching-connected”) are stated informally.
• Self-supervised vs supervised (Fig. 2B, §5; Fig. 6, App. B.2.2). SwAV and DINO bottoms into a supervised ResNet-50: penalty about 0–3 points at all layers. SimCLR: rising to about 8.5 near the top; SimCLR’s own accuracy is 68.8% against about 75% for the others, so near the top this is mostly the gap between the two base models. On CIFAR-10, SimCLR and supervised ResNet-18s stitch in both directions within about 3 points, the same range as two supervised seeds, while their CKA falls to 0.35 at layer 2.
• Label distribution (Fig. 3A). Bottoms trained on Object vs Animal labels or with 10% or 50% label noise, into a standard top: within about 3 points up to the middle of the network, then about 12–23 points at the top. With 100% random labels: about 8 points at a quarter of the depth, 55 at about 0.35, 80 at the top.
• Samples (Fig. 2C). Top trained on 10K CIFAR-10 samples (error 33.66%). A 25K bottom lowers the error from about 15% of the depth on, by about 5 points at mid depth and about 22 at the top. A 5K bottom raises it by about 15–18 points from 40% of the depth on. The 10K bottom itself, which should be the baseline, wanders from +2 to about −7 points; the paper does not say whether it is the top network’s own layers or a second 10K network.
• Training time (Fig. 3B). Top at epoch 80 (error 8.59%). The epoch-160 bottom is about +0.5 points at shallow layers and about −0.3 to −0.8 only from 60% of the depth on. The epoch-40 bottom rises to about +4.7 at the top. The text’s “representations improve in a manner that is compatible” rests on that last-layer gap of under one point.
• Width (Fig. 3C, §6). Top at 0.25× width (error 8.56%). The text lists {0.25×, 1×, 2×}; the figure also has 0.125×. The 2× bottom is about +0.5 to +1 points through 60% of the depth and about −0.8 only after that. The 1× bottom is worse than the 0.25× self-stitch at mid depth (about +2 to +3). “Can be stitched to those with lower width and improve performance, but not vice versa”: the reverse direction is not plotted.
• Kernel size (App. B.1, Fig. 4). Three unlabelled curves of stitched test error against kernel sizes 1–9; one gets worse with larger kernels. The text calls the effect “minimal”.
• Fine-tuning (App. B.3, Fig. 8). With a random ResNet-164 bottom, retraining the whole top gives a lower penalty (about 9.5–15) than stitching (about 14–18), so fine-tuning flatters a random representation more. The figure is captioned “More is similar with CKA?”, a copy of Fig. 7’s caption, and its x-axis, labelled “fraction”, runs 5 to 25.
• Freeze training (§6, App. B.4, Fig. 9). Layers whose penalty is near zero at 5K samples are frozen and the rest trained on 50K: final error about 0.11 against about 0.08 for full training. The main text says “the first three, and the last three layers … within 2%”; the appendix says layers {0, 1} and {8, 9, 11}, five layers, and “up to ≈ 3%”. It also refers to a top trained on the full dataset in Fig. 2C, whose top was trained on 10K.

Claims

id	claim	strength	support
C1	Stitching penalty is a better tool than CKA for comparing representations	argument, with examples	§3: the spurious-coordinates example (a construction), the pixel-shuffle example (an argument), and Table 1
C2	Two networks of one architecture and data distribution, trained from different seeds, are stitching-connected	moderate (experiment), stated as an informal conjecture	Fig. 2A, one pair per architecture, six architectures, CIFAR-10 only
C3	Supervised and self-supervised networks learn representations that stitch at all layers	moderate (experiment)	Fig. 2B (ImageNet, three SSL methods) and Fig. 6 (CIFAR-10, SimCLR); single runs
C4	Early layers are insensitive to label quality; late layers are not	moderate (experiment)	Fig. 3A
C5	Representations trained with more data are better versions of the same representation	moderate for samples	Fig. 2C: 25K into 10K lowers error from shallow layers on
C6	The same holds for width and training time	weak	Fig. 3B–C: improvement under one point, only in the top 40% of layers; mid-layer penalties are positive
C7	Stitching is asymmetric: wider into narrower helps “but not vice versa”	not supported here	no figure shows the reverse direction
C8	Layers have their own sample complexity, so some can be frozen early	weak	§6 and App. B.4 disagree on which layers and how close
C9	The stitching layer is not doing the learning	moderate for deep layers, weak for shallow ones	random-bottom control, Fig. 2A–B, Fig. 5: high penalty deep, near zero in the first fifth of layers

Concepts

• stitching penalty — L_ℓ(r; A) − L(A), with L_ℓ(r; A) the least loss of A’s top layers fed r through some s ∈ S; measured with a trained s.
• stitching family S — the allowed adapters; it is where invariances are chosen (“we can explicitly ensure invariance under any given family of transformations by adding it to the stitching layer”, §3).
• stitching-connected — every stitched model S_i, i = 0, …, L, joining B’s first i layers to A’s rest, has test loss “comparable to A” (Definition 1). S_0 = A and S_L = B, so the sequence is a discrete path between the networks.
• “Anna Karenina” scenario — all successful models learn roughly the same internal representations, and better models learn better versions of them; its opposite is “snowflakes”.
• “more is better” — r trained with more resources should lower A’s error when plugged in, i.e. have a negative penalty.

Connections

• Lenc and Vedaldi (LIT-363, NOTE-309). The source of the stitching layer. NOTE-309 found that the identity stitch fails (> 99% error) and that their deep-layer equivalence is judged by recovered task accuracy through a learned layer. This paper keeps the second feature, makes it the definition, and uses a much smaller adapter (1×1 rather than m × m × D filters). Lenc and Vedaldi’s ImageNet/Places stitching found Conv5 not interchangeable; this paper finds ResNets interchangeable at every layer, on CIFAR-10 and between ImageNet training methods.
• CKA and the kernel view (THEORY-008, THEORY-002). THEORY-008 holds that what a regularised linear readout can decode is a function of the normalised kernel, and that CKA averages readout agreement. Stitching departs from that view in two ways the paper names: it is nonlinear in what it decodes (A’s whole top is the decoder), and it is tied to one task. The paper’s spurious-coordinates example, where CKA drops but the representation is unchanged in use, is the kind of case where the two part.
• Mode connectivity (THEORY-112, THEORY-116). The paper calls stitching connectivity “complementary to mode connectivity”: a discrete path whose intermediate models each share all but one layer with an endpoint. THEORY-112 says the linear barrier between seeds largely vanishes once the architecture’s symmetries are factored out. Stitching connectivity is consistent with that, and weaker: the 1×1 convolution is a general linear map per layer, a larger group than permutations, and it is fitted to the task loss, not to the weights. THEORY-116 reads a surviving barrier as a difference in mechanism; the label-noise result here, where early layers stitch and late layers do not, is the same shape of finding by another measure, and it is my connection, not the paper’s.
• Platonic representation hypothesis (LIT-302, NOTE-286). NOTE-286 lists “model stitching” first among the convergence evidence Huh et al. survey. This paper is that evidence for vision, at one architecture: it shows interchangeability up to a learned linear map for one task, not convergence of kernels.
• Concurrent work. The camera-ready names Csiszárik et al. (NeurIPS 2021), which varies the stitching layer, including a sparsity penalty (not read). It cites it as NeurIPS volume 35; the 2021 volume is 34.

Bearing on the record

• It produces THEORY-176. The paper’s “more is better” is an ordering of representations, and the record holds Blackwell’s order of experiments (THEORY-156). The two are different orders. Taking the representation r(X) as an experiment about the label Y:
  • the penalty is one decision problem, the task’s loss, where Blackwell’s order quantifies over all of them;
  • the decision rules allowed are A_{>ℓ} ∘ s with s ∈ S, not every rule, so the stitched loss bounds r’s best attainable risk from above and L(A) bounds that of A’s layers from above, and comparing two upper bounds orders neither best risk;
  • representations that are Blackwell-equivalent, each a function of the other, can stitch differently. The paper wants this: its pixel-shuffle argument (§3) says a 1×1 stitcher “can predict that shuffling pixels leads to a ‘worse’ representation”, though a shuffle loses nothing;
  • the one point where they meet is exact reproduction: if some s ∈ S had s ∘ r = A_{≤ℓ}, A’s layers would be a function of r, so r would dominate them in Blackwell’s order and the penalty would be zero. A negative penalty, the paper’s evidence for “more is better”, therefore shows that s ∘ r did not reproduce A_{≤ℓ}. These steps are my derivations from the paper’s definitions and THEORY-156, not the paper’s.
• CLAIM-125. Stitching is a learned, directed transport between representations, judged by what a fixed downstream decision-maker can do with it. That is the shape of the comparison CLAIM-125 calls open. The paper does not settle it: its judgement is by one decision problem, through a fixed head, with the map trained on the problem itself, and it gives no characterisation, only measurements. It is prior art for the practice of comparing representations by decision performance, and it does not meet that claim’s defeated_if.
• CLAIM-050. It is an ML counterpart of “fidelity relative to the decisions the receiver must make”, for a single decision. It also shows what is lost by fixing the decision: Blackwell-equivalent representations can be judged different.
• CLAIM-106. Its penalty is graded and directional, and equivalence (zero penalty both ways) is a special case, which is that claim’s structure in a different domain. The paper’s evidence for direction is thin (C7).
• ML practice. It recommends stitching as a diagnostic (“we hope that model stitching will become a part of the standard diagnostic repertoire of the deep learning community”) and suggests freezing layers early as a training speed-up. Both are practice; the anthology is the place for them, hence anthology-candidate.

Limitations

• No theory. “Formal” in the paper means a definition. Definition 1 and Conjecture 2 are informal, and no property of the penalty is proved: not that it is a preorder, not transitivity, not invariance under any group.
• Single runs, values from plots. No seeds, no error bars, no tables. Differences of a point or two, which carry the width and training-time claims, are within what the 10K self-stitch baseline wanders (Fig. 2C).
• Mostly one architecture and one dataset. ResNet-18 on CIFAR-10 for everything in §§5–6 except the ImageNet SSL comparison. Vision only, as the authors say.
• The comparison is relative to a fixed top. Every “more is better” comparison is r against A’s own layers through A’s head. Chains such as 25K > 10K > 5K are each against the 10K head; whether the order changes with the head is not tested.
• The endpoints are trivial. Near the top of the network the stitched model is nearly the bottom network, so the penalty there is close to the difference between the two base models’ errors. The informative points are the middle layers, where for width and training time the sign goes the other way.
• The random-network control fails at shallow layers. A random bottom stitches with near-zero penalty in the first fifth of the layers (the paper reads this as early layers being near-random), so low early-layer penalties carry little information about learned structure.
• The sign slip in §2 and the B.4/§6 and Fig. 8 inconsistencies noted above.

Open questions

• Is the stitching order, for a fixed task, a preorder, and does it depend on the choice of top network? A test across several heads would say.
• Does stitching connectivity survive restriction of S to the architecture’s symmetry group (permutations, or orthogonal maps for a transformer’s residual stream)? That would connect it to THEORY-112’s alignment results.
• If stitching is repeated over a family of tasks on one input distribution, does the resulting order approach Blackwell’s order on some latent variable? Nothing here bears on it, since every comparison is one task.