Truncated singular value decomposition minimizes squared pixel error, but selecting a different subset of the same singular components can lower spatial-gradient error. For valid forward differences, we prove that both residual errors are sums of contributions from the omitted components. Consequently, a normalized pixel–gradient objective is minimized by retaining the highest-scoring components under a fixed total budget. A two-component example shows that gradient-only selection can discard the constant intensity term. We compare four rules at five budgets on twelve photographic crops (240 reconstructions), with a separate 84-setting weight sweep and 600 exhaustive small-matrix checks. At an equivalent rank of 32 per channel, equal weighting reduces mean normalized gradient error by 0.685% relative to pixel-optimal selection, while mean normalized pixel error increases by 4.155% and mean paired PSNR decreases by 0.208 dB. On the coffee crop, gradient-only selection loses 19.723 dB of PSNR for a 1.768% gradient-error reduction. The sorting guarantee holds for the fixed, unclipped singular dictionary; the measurements show that small gradient gains may carry appreciable pixel costs.
