Neural Spectral Capacity: Measuring and Designing Architectures from Network Specification Alone

Predicting relative trained performance from architectural specifications can guide design when training costs limit experimentation. Neural Spectral Capacity (NSC) provides such a heuristic, but scores projections independently and assigns identical scores to compatible reorderings of the same layer configurations. We introduce Residual-Aware Neural Spectral Capacity (ResNSC) for Transformers, using a specification-derived statistical reference for accumulated residual updates to determine the input and residual-state statistics in each projection's capacity calculation, while retaining additive evaluation without data or instantiated weights. When residual widths vary, earlier choices affect later scores, so we augment NSC's allocator (NSC-DP) with residual history beyond its layer-and-budget state. ResNSC-DP jointly allocates residual and feed-forward widths under a resource budget, with certified bounds on the best numerically evaluated ResNSC score within a specified finite candidate set. At the 121M and 374M pretraining scales, selected architectures achieve lower mean perplexity than uniform, variable-width, and tapered allocations on FineWeb-Edu, WikiText-2, and LAMBADA at comparable parameter budgets. In inherited-weight LLaMA-7B pruning using the LoNAS supernet, ResNSC-DP improves eight-task average accuracy over NSC by 3.28--5.46 percentage points across eight matched parameter budgets. Paired reversals preserve the projection shapes and NSC scores yet worsen performance in both settings, exposing performance differences hidden by position-independent scoring. The code for this work is available at https://github.com/Optima-CityU/neural-spectral-capacity.

Scaling laws have made language-model performance predictable from model size, data, and compute, but they typically treat the optimizer as a fixed training detail. We show that this assumption misses a fundamental axis of representation scaling: how effectively the optimizer converts added FFN width into utilized spectral capacity. Using eigenspectra of feed-forward network representations, measured through soft and hard spectral-ranks, we find that the same Transformer architecture realizes markedly different spectral scaling laws when trained with different optimizers. Holding architecture and width schedule fixed, AdamW exhibits weak hard-rank scaling ($β$=0.44) on rare-token (TAIL) representations where learning is known to be hardest, whereas Muon achieves linear scaling ($β$=1.02) in the same regimes, a $2.3\times$ increase in the scaling exponent. This difference is not reducible to validation loss: AdamW configurations can match low-rank Dion variants in perplexity, under extended training, while exhibiting sharply different spectral geometry, demonstrating that matched loss does not imply matched representation structure. Hard--soft rank asymmetry further reveals that optimizers differ not only in how much capacity is realized, but also in how that capacity is structured across eigenmodes. To disentangle optimizer effects from architectural ones, we compare against architectural interventions (e.g., attention rank and positional encoding), and find that optimizer-induced spectral shifts often exceed the architectural effects. These results suggest optimization as a first-class axis of representation scaling, motivating optimizer--architecture co-design.

Transformer architectures achieve state-of-the-art performance across a wide range of pattern recognition and natural language processing tasks, but their scaling is accompanied by substantial parameter growth and redundancy in the embedding dimension. In this work, we introduce a structured spectral factorization of the embedding space based on the L-product for third-order tensors. By reshaping token representations into spectral tensor slices and performing attention and feed-forward operations in the transform domain, we obtain a Tensor Transformer architecture that decomposes the encoder into p independent spectral sub-transformers while preserving standard Transformer semantics. We prove that the proposed L-Transformer is spectrally equivalent to p parallel Transformers operating on reduceddimensional embeddings, which yields approximately 1/p reduction (up to lower-order terms such as biases and normalization parameters) in encoder parameters under fixed total embedding size. When instantiated with a real-valued Discrete Cosine Transform (DCT), the method remains fully differentiable and compatible with existing training pipelines. Beyond compression, the spectral decomposition introduces an inductive bias over embedding frequencies, enabling slice-dependent frequency scaling that improves generalization. Experiments on IMDB and AGNews show that the proposed model can substantially reduce encoder parameters (up to 75% for p=4) while maintaining competitive accuracy. On IMDB, the tensorized encoder matches or improves upon the standard baseline under compression, whereas on AGNews at moderate width we observe a small accuracy decrease in exchange for a 4 times encoder reduction; at BERT-base width (d=768), performance returns to parity.