Inverse compositional spatial transformer network

⚠️ Curated Limitations & Negative Results

• Absorbing inversion into a predictor is a learned parameterization choice, not proof that every output is the exact inverse of a separately computed forward update. The affine representation alone does not enforce nonsingularity. Inverse-warp semantics require an invertible, suitably conditioned linear part, and homography sampling requires nonzero homogeneous denominators over the sampled domain. Preserve the source’s literal M(delta p) M(p_in) convention; do not silently add another inverse, reverse multiplication or replace composition with parameter addition.
• The full recurrent state derivative includes the feedback route C_delta Df D_p[warp(I,p)] in addition to C_p. A bounded hand example uses the finite grid x,y in {0,1,2} with intensity I(x,y)=x, sampling at (1/2,1/2) under horizontal translation p with abs(p)<1/4. Bilinear sampling gives 1/2+p, and an affine predictor delta=1/2−sample=−p yields composed translation zero: both direct composition partials are one while the total derivative is zero. This demonstrates nonimplication of the source’s residual-gradient argument, not an implementation error or a failure claim about a trained model.
• Even fixed small increments provide no uniform guarantee against vanishing, exploding or ill-conditioned long products. For 0<epsilon<1, isotropic incremental scales 1−epsilon and 1+epsilon yield direct magnitudes (1−epsilon)^K and (1+epsilon)^K but condition number one; an anisotropic diagonal pair yields condition ratio ((1+epsilon)/(1−epsilon))^K. These hand constructions separate gradient magnitude from conditioning and do not assert the behavior of every image sampler or task-loss gradient.
• Repeatedly sampling the retained original image avoids replacing it with successive cropped or interpolated outputs, but it cannot reconstruct pixels outside the original support, true occlusion or lost resolution. Predictor sampling still occurs. Interpolation knots, boundaries, padding, finite precision and projective conditioning require implementation conventions; the architecture does not certify smoothness or exact raster preservation everywhere.
• Sharing a predictor does not make its weights a classical template-Jacobian pseudoinverse, establish a learned scalar potential, or prove monotone descent or convergence. The paper’s Figure 7 reports additional-iteration improvements under its planar alignment protocol; it does not certify arbitrary extrapolation beyond the trained unrolling. Parameter sharing does not make additional predictor, sampler and composition steps cost-free. The finite alignment and classification comparisons establish no universal accuracy, memory or runtime advantage.
• Evidence is the exact nine-page official CVPR 2017 proceedings PDF. The acquired CVF page numbering and Crossref IEEE numbering differ and are explicitly separated; no exact publication day or preprint byte identity is asserted. The official supplementary ZIP returned HTTP 404, so no supplementary content is claimed. Historical source inspection did not include author code or full original LK, IC-LK and SDM papers, and did not execute training or reproduce benchmarks. No implementation-level coordinate normalization, padding, device behavior or numerical accuracy is certified.