Compositional spatial transformer

⚠️ Curated Limitations & Negative Results

• The affine coordinate convention and multiplication order are part of the stated mechanism. Composing singular affine matrices is algebraically possible, but interpreting a warp as invertible requires a nonsingular, suitably conditioned linear part. Equation 9 imposes no determinant constraint by itself. The homography experiments additionally require nonzero homogeneous denominators on the sampled domain; the affine Equation 11 does not specify homography derivatives. An incremental inverse, when required by a chosen interpretation, is not obtained by simply negating all six parameters.
• Retaining the original image prevents information loss caused by repeatedly replacing it with intermediate cropped or interpolated outputs. It does not recover pixels absent from the original source, physical occlusion or lost resolution. Intermediate samples for the predictors still occur. Finite grids, out-of-range sampling, interpolation, aliasing and numerical error remain relevant; exact matrix composition does not imply exact raster preservation.
• Equation 11 contains correct composition partials. If C(p,delta) is the composition and delta=f(warp(I,p)), the full state derivative is C_p+C_delta Df D_p[warp(I,p)] where differentiable. The identity term in a partial alone gives no arbitrary-horizon gradient guarantee. For 0<epsilon<1, constant incremental linear parts (1−epsilon)I or (1+epsilon)I produce direct gradient magnitudes (1−epsilon)^K or (1+epsilon)^K; these isotropic examples still have condition number one. Anisotropic scales can instead worsen conditioning. These are bounded hand constructions, not reports about trained-model failure or author-code backpropagation.
• Differentiable or subdifferentiable sampling enables optimization but does not imply smoothness at all interpolation knots and boundaries. The displayed composition partials must be combined with the predictor and sampler derivatives and their domain conventions. No padding, subgradient, coordinate-range or singular-transform handling is verified at implementation level.
• A learned linear predictor can share the algebraic form of a classical alignment regressor without equaling a particular template-Jacobian pseudoinverse or minimizing the same objective. The source’s LK/SDM analogy, residual-gradient explanation and selected empirical results do not establish universal solver equivalence, monotone descent, convergence or accuracy and computational advantages. This record denotes the Figure 3 reusable module; cascade usage does not silently change its identity into a whole classifier or require untied weights.
• 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.