Source
Kokhlikyan, Miglani, Alsallakh, Martin and Reblitz-Richardson (2021), LIT-724, §2–3 and Appendix A; with Adebayo et al. (2018), LIT-713, which states the conjecture and declines to measure it.
The account
Global Integrated Gradients along input dimension i is
Φ(F, xⁱ) = (xⁱ − x₀ⁱ) · ∫₀¹ ∂F(x₀ⁱ + α(xⁱ − x₀ⁱ))/∂xⁱ dα
The integral depends on the model. The leading factor (xⁱ − x₀ⁱ) does
not. With the usual zero or black baseline it is essentially the input. So
the map is a model-dependent quantity multiplied, pixel by pixel, by a
model-independent picture of the input — and a similarity metric that reads
spatial structure will report that the map is unchanged when the model is
destroyed, because the part of it carrying the structure never depended on the
model in the first place.
It was written down informally in 2017, in a visualization section.
LIT-731 §3.1, on whether to multiply a gradient map by the input:
the multiplication “does tend to produce visually simpler and sharper images,
although it can be unclear how much of this can be attributed to sharpness in the
original image itself. For example, a black/white edge in the input can lead to
an edge-like structure on the final visualization even if the underlying
sensitivity map has no edges.” That is this account, a year before LIT-713
suspected it and four before LIT-725 measured it — filed as advice about colour
scales rather than as a claim about faithfulness, which is why the record rebuilt
it from the 2021 measurement.
The same paper adds a structural objection the record had from nobody: under gradient⊙input, “pixels with values of 0 will never show up on the sensitivity map”, so a correctly classified black ball on a white background is never highlighted if black encodes zero.
The same factor appears in InputXGradient, DeepLift and Gradient SHAP, which is
why they land in the same column of LIT-713’s verdicts. Guided Backprop and
Guided GradCAM are outside this account: they modify gradients during
back-propagation and fail for a separate reason.
The boundary, now that a second account is held. The cluster’s failures partition three ways, and this document owns the first row:
| family | mechanism | held as |
|---|---|---|
| Integrated Gradients, InputXGradient, DeepLIFT, Gradient SHAP | this account — the model-independent input multiplier | here |
| DTD, LRP-α1β0, Excitation BP, PatternAttribution | rank-1 convergence of a non-negative chain | THEORY-114 |
| Guided Backprop, Deconv, RectGrad | partial image recovery | THEORY-115 |
LIT-729 supports this account from outside its own subject: methods that “rely on the gradient directly (Smilkov et al. 2017; Sundararajan et al. 2017) … [do] not converge”, so Integrated Gradients’ randomization behaviour is not the rank-1 mechanism, which is what leaves the multiplier as the explanation. DeepLIFT sits in the first row and outside the second, which is not a contradiction: it carries an input multiplier and keeps negative contributions, so it has one of the two failure modes and escapes the other.
Why this is an account rather than a restatement
LIT-713 already suspected the multiplier. What it says is that parameter and
data invariance “is likely to be related to the input multiplier in some
gradient-based methods; however, they do not measure input multiplier’s
quantitative impact”. A suspicion that names the right factor and runs no arm
to isolate it is a hypothesis, and the record’s standard is that a mechanism
claim earns its status when the experiment includes an arm the mechanism says
must behave differently.
There are two such arms here, and they point in opposite directions:
- Remove the multiplier, holding everything else fixed. Local IG on Inception under cascading parameter randomization: the maps change, and SSIM for the local variant is about half the global variant’s. If the insensitivity were a property of gradient attribution, dropping a model-independent scale factor would not fix it.
- Destroy the multiplier’s structure without removing it. On text, token scores are formed by summing over embedding dimensions. The account predicts that a multiplier whose spatial structure is summed away stops protecting the map — and on BERT/SST-2 the global variant becomes parameter-sensitive, with cosine-similarity medians near zero. Global IG fails the test on images and passes it on text, which a “gradient attributions are insensitive to weights” reading cannot accommodate at all.
The second arm is the one that makes this worth filing. An account that only explained the image result would be indistinguishable from the label “multiplier”; this one predicts a sign change across modality and gets it.
What it does not say
- It is not a defence of Integrated Gradients. The account says the
randomization test’s verdict is about the multiplier. It says nothing about
whether the local variant is faithful —
LIT-713’s tests are necessary conditions, and passing one is not a certificate. Every caution SOTA-430 carries about that still holds. - It does not explain the metric split. On the same Inception experiment, Spearman rank correlation drops to near zero for both variants while SSIM separates them. So SSIM is reading the structure this account is about and Spearman is not, which is consistent with it but is a fact about the metrics that no arm here isolates.
- It does not cover the smoothness confound. Randomization also degrades the integral approximation — infidelity rising from 2.84 to 1.27 × 10⁷ down the Inception stack — and that is a separate mechanism, diagnosed in the same paper and not separated from this one.
- Two architectures, one group, and that group maintains captum, whose API is where you switch the multiplier off. The measurement is a real control; the choice of which control to run is not disinterested.