Tensor Decomposition

Canonical Polyadic (CP) decomposition factorizes an N-way tensor into a sum of component rank-one tensors. For a 3D convolutional kernel, this replaces one large tensor with three small factor matrices, reducing parameters from O(XYZ) to O(R(X+Y+Z)) where R is the tensor rank. This is particularly effective for compressing depthwise separable convolutions in neural receiver architectures, where spatial and channel correlations can be exploited independently.

Tucker decomposition represents a tensor as a small core tensor multiplied by factor matrices along each mode. Unlike CP, Tucker allows different ranks along each dimension, providing more flexibility for compressing multi-dimensional signal processing tensors. In RF applications, this is used to compress 4D weight tensors (output channels, input channels, kernel height, kernel width) by applying mode-specific compression ratios that preserve frequency-domain feature extraction fidelity.

Tensor Train (TT) decomposition represents a high-dimensional tensor as a chain of low-rank 3D core tensors, reducing exponential storage complexity to linear in the number of dimensions. This is critical for compressing fully-connected layers in transformer-based signal classifiers where weight matrices can be reshaped into higher-order tensors. TT-format enables extreme compression ratios exceeding 100x while maintaining automatic modulation classification accuracy.

Singular Value Decomposition (SVD) is applied to 2D weight matrices to retain only the top-K singular values and corresponding vectors. For a weight matrix W, this approximates W ≈ U_k Σ_k V_k^T, reducing parameters from MN to K(M+N+1). In neural channel estimation models, SVD-based low-rank approximation compresses fully-connected layers by 75-90% with negligible mean squared error increase in channel state information reconstruction.

Decomposed layers require recovery fine-tuning to restore accuracy lost during factorization. This process involves:

• Freezing decomposed factor tensors initially
• Training with reduced learning rates (1e-4 to 1e-5)
• Using knowledge distillation from the original uncompressed model
• Monitoring IQ constellation fidelity as a domain-specific metric Typically, 5-20 epochs recover 95-99% of original accuracy for RF classification tasks.

Optimal tensor rank is determined by the target hardware's memory bandwidth and compute architecture. Lower ranks reduce SRAM footprint but may increase reconstruction overhead. For Cortex-M7 microcontrollers running CMSIS-NN, rank selection must balance:

• Decomposition compression ratio
• Matrix multiplication overhead from factor tensors
• Cache line alignment requirements Automated rank search using Bayesian optimization finds Pareto-optimal configurations for specific NPU and MCU targets.

The foundational mathematical operation behind most tensor decomposition methods. A weight tensor W is approximated as the product of two or more smaller factor matrices, dramatically reducing the parameter count. For a convolutional layer with a 4D kernel tensor, canonical polyadic (CP) decomposition or Tucker decomposition can achieve 3-10x compression with minimal accuracy loss. The rank selection trades off between compression ratio and representational capacity.

A complementary sparsification technique that removes individual low-magnitude connections rather than restructuring the tensor. Unstructured pruning zeros out weights below a threshold, while structured pruning removes entire channels or filters. Often combined with tensor decomposition in a deep compression pipeline: decompose first to reduce the parameter space, then prune the factor matrices for additional gains. N:M sparsity patterns (e.g., 2:4) provide hardware-friendly acceleration on GPU tensor cores.

Simulates low-precision inference during the forward pass so the model learns parameters robust to quantization error. After tensor decomposition produces smaller factor matrices, QAT ensures those compact representations survive INT8 or INT4 quantization without catastrophic accuracy degradation. The straight-through estimator (STE) approximates gradients through the non-differentiable rounding operation, enabling end-to-end training of quantized decomposed models.

Trains a compact student model to replicate the soft output distribution of a larger teacher model. When combined with tensor decomposition, the teacher's high-capacity representations guide the student's factorized layers. The student learns not just the correct classification but the teacher's dark knowledge—the relative probabilities across all classes. This is particularly effective for RF modulation classification where subtle signal distinctions are easily lost during aggressive decomposition.

A factorized convolution that decomposes standard spatial filtering into two steps: a depthwise convolution applying a single filter per input channel, followed by a pointwise convolution (1x1) combining channel outputs. This is essentially a constrained form of tensor decomposition built directly into the architecture. Reduces computation by 8-9x compared to standard convolutions and forms the backbone of MobileNet architectures widely used in on-device RF processing.

A graph optimization that mathematically absorbs batch normalization parameters into the preceding convolutional layer's weights and biases. After tensor decomposition restructures weight tensors, folding eliminates the separate BN operations at inference time. The folded weights are computed as: W_folded = (γ/σ) × W_original and b_folded = β - (γμ/σ). This reduces memory accesses and eliminates runtime variance calculations critical for deterministic edge inference.