Compressed CP-Conv: Convolutional Learning from Factorized High-Dimensional Inputs

Convolutional neural networks conventionally operate on explicitly materialized tensors, making high-resolution and high-dimensional inputs increasingly costly or infeasible to process. We introduce *CP-Conv*, a convolutional operator that applies an arbitrary dense, trainable kernel directly to inputs represented in Canonical Polyadic (CP) form, without reconstructing the dense tensor or factorizing the convolutional kernel. For the tensor represented by the supplied CP factors, CP-Conv is algebraically equivalent to dense convolution and therefore introduces no additional approximation of the operator. Bnecause its output is generally dense, we further introduce *Compressed CP-Conv* (CCP-Conv), which fuses convolution with mode-wise output compression and directly produces a compact representation suitable for standard neural-network layers. We derive the computational and storage complexity of both operators and verify numerical agreement with their dense counterparts. Controlled benchmarks yield operator-level speedups of up to $3.9\times 10^3$. Across supervised experiments on synthetic data, UCF101, Ego4D, high-dimensional Fokker–Planck problems, and nonlinear shallow-water dynamics from PDEBench, CCP-Conv closely matches matched dense baselines when dense processing is feasible, while substantially reducing training cost. These experiments include both inputs available directly in CP form and dense data represented through approximate CP decomposition. CCP-Conv also enables convolutional learning on inputs whose implicit dense representation contains up to $2.94\times 10^{11}$ values, well beyond the scale that can be processed densely within the available resources. CCP-Conv integrates into a pretrained R3D-18 architecture with comparable classification performance and supports unsupervised representation learning without dense reconstruction. Together, these results show that exploiting structure directly in the input representation can extend dense-kernel convolutional learning to regimes where materializing the input tensor is computationally prohibitive. Reference implementations in JAX and PyTorch, together with reproduction scripts, are available in the supplementary material.