{
  "paper_id": "cara",
  "paper_title": "Canonical Rank Adaptation: An Efficient Fine-Tuning Strategy for Vision Transformers",
  "version": "v2",
  "status": "pending_review",
  "review_notes": "v2: Removed D1-005 (→D2-009) and D1-018 (→D2-011) per R2 review near-duplicate findings. Trimmed all D1 claims to match benchmark granularity (~150 chars avg, down from 209). D2/D3/D4 unchanged. R2 review: no HARD EXCLUDE violations, no completeness gaps.",
  "D1": [
    {
      "id": "cara-D1-001",
      "claim": "ViT tensorisation shapes: MHA tensor W^mha ∈ R^{3l × d_model × h × d_h} (Q/K/V across l blocks, h heads); FFN tensor W^ffn ∈ R^{9l × d_model × d_model} (W^O/W^up/W^down across l blocks, d_ff=4·d_model).",
      "source": "Section 3.2.1"
    },
    {
      "id": "cara-D1-002",
      "claim": "Individual head dimension d_h = d_k = d_v; feed-forward expansion ratio d_ff = 4·d_model (standard ViT).",
      "source": "Section 3.2.1"
    },
    {
      "id": "cara-D1-003",
      "claim": "MHA CPD factor shapes: A^(1)∈R^{3l×R}, A^(2)∈R^{d_model×R}, A^(3)∈R^{h×R}, A^(4)∈R^{d_h×R}, λ^A∈R^R.",
      "source": "Section 3.2.2"
    },
    {
      "id": "cara-D1-004",
      "claim": "FFN CPD factor shapes: B^(1)∈R^{9l×R}, B^(2)∈R^{d_model×R}, B^(3)∈R^{d_model×R}, λ^B∈R^R.",
      "source": "Section 3.2.2"
    },
    {
      "id": "cara-D1-005",
      "claim": "CaRA weight update: W + α·δW, α is scaling hyperparameter; δW merged into frozen weights at inference for zero overhead.",
      "source": "Section 3.2.2, Equations 9, 11"
    },
    {
      "id": "cara-D1-006",
      "claim": "VTAB-1k setup: ViT-B/16 pretrained on ImageNet-21k; 19 datasets across Natural (7), Specialized (4), Structured (8); 1000 samples/dataset, 80-20 train/val split; top-1 accuracy over 10 runs.",
      "source": "Section 4.1, Appendix C.2"
    },
    {
      "id": "cara-D1-007",
      "claim": "VTAB-1k per-dataset hyperparameter ranges (Table 8): R ∈ {16,32}; α ∈ {0.1,1,2.5,10,50,100}; λ_μ ∈ {0.9,1.0,1.08,1.16,1.2,1.3,1.35,1.5}; λ_σ ∈ {0.0,0.01,0.02,0.028,0.03,0.05,0.06,0.07,0.1}.",
      "source": "Section C.2, Table 8"
    },
    {
      "id": "cara-D1-008",
      "claim": "FGVC datasets: CUB-200-2011 (200 classes), NABirds (555), Oxford Flowers (102), Stanford Dogs (120), Stanford Cars (196); full training set, val split seed=0.",
      "source": "Section 4.2, Appendix C.3"
    },
    {
      "id": "cara-D1-009",
      "claim": "FGVC hyperparameters (Table 9): R=32 fixed; α ∈ {0.001,0.01,10,100}; λ_μ ∈ {0.9,1.0}; λ_σ ∈ {0.0,0.02}.",
      "source": "Section C.3, Table 9"
    },
    {
      "id": "cara-D1-010",
      "claim": "ViT-L few-shot setup: ImageNet-21k pretrained ViT-L; CIFAR100, Food101, Flowers102, Resisc45; 10 examples/class (numpy seed=6); full FT params=303.3M.",
      "source": "Section 4.3, Appendix C.4"
    },
    {
      "id": "cara-D1-011",
      "claim": "ViT-L hyperparameters (Table 10): R=32, α=0.01 fixed; λ_μ=1.0; λ_σ ∈ {0.0,0.01}; 5 runs per dataset.",
      "source": "Section C.4, Table 10"
    },
    {
      "id": "cara-D1-012",
      "claim": "ViT-L baseline configs: DoRA and PiSSA at R=8, α=8, 100 iterations via HuggingFace PEFT.",
      "source": "Section C.4"
    },
    {
      "id": "cara-D1-013",
      "claim": "Hardware: 1× Nvidia GA100 (VTAB-1k), 1× Nvidia H100 (FGVC), Nvidia RTX A5000 (eval), up to 8× GA100 (NLU).",
      "source": "Appendix C.1"
    },
    {
      "id": "cara-D1-014",
      "claim": "Software: PyTorch autograd, Tensorly for tensor ops, timm for pretrained checkpoints.",
      "source": "Appendix C.1"
    },
    {
      "id": "cara-D1-015",
      "claim": "CaRA trainable parameter counts: ~0.06M (VTAB-1k avg ViT-B), 0.08M (FGVC ViT-B), 0.076M (ViT-L), 0.134M (Swin-B), 0.055M (RoBERTa-Base R=64, ~0.04% of full FT).",
      "source": "Sections 4.1, 4.2, 4.3, D.1, D.2"
    },
    {
      "id": "cara-D1-016",
      "claim": "Rank ablation design: CaRA vs FacT-TT/FacT-TK on VTAB-1k Specialized group, ranks varied as powers of two.",
      "source": "Section 5.1"
    },
    {
      "id": "cara-D1-017",
      "claim": "Dimension ablation tensor variants (Table 5, R=32): Wmha-5D (3×l×d_model×h×d_h), Wmha-3D (3l×d_model×h*d_h), Wmha-4D/CaRA (3l×d_model×h×d_h), merged MHA+FFN (12l×d_model×d_model).",
      "source": "Section 5.2, Table 5"
    },
    {
      "id": "cara-D1-018",
      "claim": "GLUE config: RoBERTa-Base (125M full FT params), CaRA on Q/V MHA matrices only, R=64, α=100, all λ init=1.",
      "source": "Section D.2"
    },
    {
      "id": "cara-D1-019",
      "claim": "Saliency analysis setup: FGVC-Aircraft (70 aircraft families), ViT-B/16, CaRA at R=32 and R=16, Integrated Gradients.",
      "source": "Section 5.3, Appendix E"
    }
  ],
  "D2": [
    {
      "id": "cara-D2-001",
      "claim": "Canonical-Polyadic Decomposition (CPD) of a fourth-order tensor T in R^{I×J×K×L}: T ≈ Σ_{r=1}^R λ_r^S · s_r^(1) ∘ s_r^(2) ∘ s_r^(3) ∘ s_r^(4), where ∘ denotes outer product, R is tensor rank, λ^S in R^R is the scaling vector, and S^(1) in R^{I×R} through S^(4) in R^{L×R} are the CP-factor matrices. The CPD represents I·J·K·L elements using only R + R·(I+J+K+L) parameters.",
      "source": "Section 3.1.1, Equation 1"
    },
    {
      "id": "cara-D2-002",
      "claim": "LoRA weight update for a linear layer with frozen pre-trained weight W in R^{n×m}: y = W x + α · B A^T x, where B in R^{n×R} and A in R^{m×R} are trainable low-rank matrices, α is a scaling hyper-parameter, R << min(n, m). A initialized with random Gaussian (Glorot), B initialized to zeros so δW = B A^T is zero at training start. During inference, δW is merged into W with no additional latency. Trainable parameters: (n + m) · R.",
      "source": "Section 3.1.2, Equation 2"
    },
    {
      "id": "cara-D2-003",
      "claim": "MHA tensorisation constructs W^mha in R^{3l × d_model × h × d_h} in three steps: (1) Per-head stacking — for each head i, form E_i = [W_i^Q, W_i^K, W_i^V] in R^{3 × d_model × d_h} by stacking head-specific slices of Q/K/V weight matrices; (2) Cross-head stacking — for block j, form L_j = [E_1, ..., E_h] in R^{3 × d_model × h × d_h}; (3) Cross-block stacking — form [L_1, ..., L_l] in R^{3 × l × d_model × h × d_h}, then merge first two dimensions to obtain W^mha in R^{3l × d_model × h × d_h}.",
      "source": "Section 3.2.1, Equations 3-5"
    },
    {
      "id": "cara-D2-004",
      "claim": "FFN tensorisation constructs W^ffn in R^{9l × d_model × d_model} in three steps: (1) Reshape W^O in R^{d_model × d_model} to R^{1 × d_model × d_model}, W^up in R^{d_model × d_ff} to R^{4 × d_model × d_model}, W^down in R^{d_ff × d_model} to R^{4 × d_model × d_model} (since d_ff=4·d_model); (2) Per-block — for block j, form F_j = [W^O_tens, W^up_tens, W^down_tens] in R^{9 × d_model × d_model}; (3) Cross-block — stack F_j across l blocks to obtain W^ffn in R^{9l × d_model × d_model}.",
      "source": "Section 3.2.1, Equations 6-7"
    },
    {
      "id": "cara-D2-005",
      "claim": "CaRA MHA low-rank update uses CPD with four factor matrices: δW^mha = Σ_{r=1}^R λ_r^A · a_r^(1) ∘ a_r^(2) ∘ a_r^(3) ∘ a_r^(4), where A^(1) in R^{3l×R}, A^(2) in R^{d_model×R}, A^(3) in R^{h×R}, A^(4) in R^{d_h×R} are factor matrices, λ^A in R^R is the learned scaling vector. Fine-tuned weight: W^mha + α · δW^mha, with δW^mha reconstructed from CPD factors and merged for zero inference overhead.",
      "source": "Section 3.2.2, Equations 8-9"
    },
    {
      "id": "cara-D2-006",
      "claim": "CaRA FFN low-rank update uses CPD with three factor matrices: δW^ffn = Σ_{r=1}^R λ_r^B · b_r^(1) ∘ b_r^(2) ∘ b_r^(3), where B^(1) in R^{9l×R}, B^(2) in R^{d_model×R}, B^(3) in R^{d_model×R} are factor matrices, λ^B in R^R is the learned scaling vector. Fine-tuned weight: W^ffn + α · δW^ffn, with δW^ffn reconstructed from CPD factors and merged for zero inference overhead.",
      "source": "Section 3.2.2, Equations 10-11"
    },
    {
      "id": "cara-D2-007",
      "claim": "MHA CPD gradients: ∇_{λ_r^A} L = a_r^(1) ∘ a_r^(2) ∘ a_r^(3) ∘ a_r^(4); ∇_{a_r^(1)} L = I^{A1} ∘ a_r^(2) ∘ a_r^(3) ∘ a_r^(4); ∇_{a_r^(2)} L = I^{A2} ∘ a_r^(1) ∘ a_r^(3) ∘ a_r^(4); ∇_{a_r^(3)} L = I^{A3} ∘ a_r^(1) ∘ a_r^(2) ∘ a_r^(4); ∇_{a_r^(4)} L = I^{A4} ∘ a_r^(1) ∘ a_r^(2) ∘ a_r^(3), where I^{A1}∈R^{3l×3l}, I^{A2}∈R^{d_model×d_model}, I^{A3}∈R^{h×h}, I^{A4}∈R^{d_h×d_h} are identity matrices. In practice, PyTorch autograd computes these analytically.",
      "source": "Section 3.3, Equation 12"
    },
    {
      "id": "cara-D2-008",
      "claim": "FFN CPD gradients: ∇_{λ_r^B} L = b_r^(1) ∘ b_r^(2) ∘ b_r^(3); ∇_{b_r^(1)} L = I^{B1} ∘ b_r^(2) ∘ b_r^(3); ∇_{b_r^(2)} L = I^{B2} ∘ b_r^(1) ∘ b_r^(3); ∇_{b_r^(3)} L = I^{B3} ∘ b_r^(1) ∘ b_r^(2), where I^{B1}∈R^{9l×9l}, I^{B2},I^{B3}∈R^{d_model×d_model}. Pattern generalizes to n-dimensional CPD: gradient of each factor replaces it with an identity matrix of its dimension.",
      "source": "Section 3.3, Equation 13"
    },
    {
      "id": "cara-D2-009",
      "claim": "CaRA factor initialization: A^(1)∈R^{3l×R} and B^(1)∈R^{9l×R} with random normal (Glorot); A^(2)∈R^{d_model×R} and B^(2)∈R^{d_model×R} to zeros (ensuring δW=0 at start); A^(3)∈R^{h×R}, A^(4)∈R^{d_h×R}, B^(3)∈R^{d_model×R} as orthogonal matrices (Saxe et al., 2014); λ^A,λ^B∈R^R sampled from N(λ_μ, λ_σ²) with dataset-specific λ_μ,λ_σ (typical: λ_μ=1.0, λ_σ=0.01). For language models (RoBERTa), all λ initialized as ones.",
      "source": "Section 3.4, Appendix B, Sections C.2-C.4"
    },
    {
      "id": "cara-D2-010",
      "claim": "CaRA total trainable parameters: MHA CPD factors require R·(3l + d_model + h + d_h + 1) params. FFN CPD factors require R·(9l + 2·d_model + 1) params. Combined: R·(12l + 3·d_model + h + d_h + 2). Compared to LoRA per MHA projection layer requiring (d_model + h·d_h)·R, CaRA distributes cost across shared factor matrices with effective per-layer contribution of (d_model + h + d_h)·R.",
      "source": "Section 3.3"
    },
    {
      "id": "cara-D2-011",
      "claim": "Swin Transformer stage-wise tensorisation: CaRA applied independently per stage due to different d_model and h. Swin-Base Stage 1 (2 layers, d_model=128, h=4): W^mha∈R^{6×128×4×32}, W^ffn∈R^{18×128×128}; Stage 2 (2 layers, d_model=256, h=8): W^mha∈R^{6×256×8×32}, W^ffn∈R^{20×256×256}; Stage 3 (18 layers, d_model=512, h=16): W^mha∈R^{54×512×16×32}, W^ffn∈R^{162×512×512}; Stage 4 (2 layers, d_model=1024, h=32): W^mha∈R^{6×1024×32×32}, W^ffn∈R^{22×1024×1024}.",
      "source": "Section D.1, Table 11"
    }
  ],
  "D3": [
    {
      "id": "cara-D3-001",
      "claim": "CaRA evaluated on VTAB-1k: 19 diverse vision datasets grouped into Natural (CIFAR-100, Caltech-101, DTD, Flowers102, Pets, SVHN, Sun397), Specialized (Resisc45, EuroSAT, Patch Camelyon, Diabetic Retinopathy), and Structured (Clevr-count, Clevr-distance, DMLab, KITTI-distance, sNORB-azimuth, sNORB-elevation); 1000 samples per dataset, 80-20 train/val split, original test set held out; ViT-B/16 pretrained on ImageNet-21k, one Nvidia GA100 GPU; metric: Top-1 accuracy (group-wise and overall mean); baselines: Linear, Full FT, Adapter-256, VPT-Shallow, VPT-Deep, AdaptFormer, SSF, RepAdapter, NOAH, LoRA, FacT-TT, FacT-TK, SPT-LoRA.",
      "source": "Section 4.1, Table 2, Appendix C.2 Table 8"
    },
    {
      "id": "cara-D3-002",
      "claim": "CaRA evaluated on FGVC benchmark: 5 datasets — CUB-200-2011 (200 classes), NABirds (555 classes), Oxford Flowers (102 classes), Stanford Dogs (120 classes), Stanford Cars (196 classes); full training set, no subsampling; ViT-B/16 pretrained on ImageNet-21k, rank 32, one Nvidia H100 GPU; metric: Top-1 accuracy; baselines: Linear, Full FT, VPT-Shallow, LoRA, AdaptFormer, Adapter, VPT-Deep, SPT-LoRA.",
      "source": "Section 4.2, Table 3, Appendix C.3 Table 9"
    },
    {
      "id": "cara-D3-003",
      "claim": "CaRA evaluated in few-shot regime: ViT-Large pretrained on ImageNet-21k, 4 datasets (CIFAR100, Food101, Flowers102, Resisc45), 10 random training examples per class (numpy seed 6); standard test sets for CIFAR100/Food101/Flowers102, remaining unsampled data for Resisc45; CaRA with rank 32, α=0.01; DoRA and PiSSA baselines with rank 8, α=8, 100 iterations via HuggingFace PEFT; metric: Top-1 accuracy; baselines: Linear, Full FT, LoRA, VeRA, PiSSA, DoRA.",
      "source": "Section 4.3, Table 4, Appendix C.4 Table 10"
    },
    {
      "id": "cara-D3-004",
      "claim": "Rank robustness ablation: CaRA, FacT-TT, and FacT-TK trained on VTAB-1k Specialized group at ranks varying as powers of two; measured: growth in trainable parameter count and mean accuracy across methods, comparing parameter efficiency vs. performance scaling with rank.",
      "source": "Section 5.1, Figure 5"
    },
    {
      "id": "cara-D3-005",
      "claim": "Tensorization dimensionality ablation: 4 tensor configurations evaluated on VTAB-1k Specialized group at rank 32 — 5D MHA (3×l×d_model×h×d_h), 3D MHA (3l×d_model×h*d_h), proposed 4D MHA (3l×d_model×h×d_h, i.e., CaRA), and merged single-tensor (12l×d_model×d_model, combining MHA+FFN); metric: accuracy and parameter count for performance-to-parameter trade-off.",
      "source": "Section 5.2, Table 5"
    },
    {
      "id": "cara-D3-006",
      "claim": "Saliency map analysis: FGVC-Aircraft dataset (70 aircraft family categories; aircraft is an ImageNet class but pretrained model cannot differentiate families); ViT-B/16 fine-tuned with CaRA at ranks 32 and 16; Integrated Gradients generates saliency maps; visual comparison assesses whether CaRA learns discriminative aircraft parts (e.g., cockpit bump on B-747, third aft engine on L-1011) despite limited trainable parameters.",
      "source": "Section 5.3, Appendix E, Figure 6"
    },
    {
      "id": "cara-D3-007",
      "claim": "Computational cost measurement: walltime (seconds) and GPU VRAM (GB) recorded during ViT-Large fine-tuning on CIFAR100; CaRA directly compared against LoRA, DoRA, FacT-TT, and FacT-TK to assess practical training overhead of CPD-based tensor formulation vs. matrix-based and other tensor-based PEFT methods.",
      "source": "Section 5.4, Table 6"
    },
    {
      "id": "cara-D3-008",
      "claim": "Factor initialization ablation: 4 strategies evaluated on SVHN (VTAB-1k) — (a) zero A^(1), orthogonal A^(3)/A^(4); (b) orthogonal A^(1), random normal A^(3)/A^(4); (c) all random normal; (d) proposed: random normal A^(1), orthogonal A^(3)/A^(4); A^(2) always zero; metric: accuracy.",
      "source": "Appendix B, Table 7"
    },
    {
      "id": "cara-D3-009",
      "claim": "Hierarchical architecture evaluation: Swin-Base pretrained on ImageNet-21k, CaRA tensorisation applied independently per stage (4 stages with varying d_model and h); all VTAB-1k datasets; metric: group-wise mean accuracy (Natural, Specialized, Structured); baselines: Full FT, Linear, BitFit, VPT-Shallow, VPT-Deep, LoRA, SPT-LoRA, FacT-TT.",
      "source": "Appendix D.1, Table 11, Table 12"
    },
    {
      "id": "cara-D3-010",
      "claim": "GLUE benchmark evaluation: RoBERTa-Base fine-tuned with CaRA applied only to query and value projection matrices in MHA, rank 64, α=100, all λ initialized to 1; 8 GLUE tasks (MNLI, QQP, MRPC, SST-2, CoLA, QNLI, RTE, STS-B); metrics: Matthew's correlation (CoLA), Pearson correlation (STS-B), accuracy (all others); baselines: Full FT, Adapter, BitFit, LoRA (rank 8), LoTR (ranks 32, 88), LoRETTA_adapter.",
      "source": "Appendix D.2, Table 13"
    }
  ],
  "D4": [
    {
      "id": "cara-D4-001",
      "claim": "The CaRA method proceeds in a fixed sequential pipeline defined in Section 3: (1) Define CPD of a fourth-order tensor as sum of R rank-one outer products with factor matrices S^(1) through S^(4) (Section 3.1.1); (2) Review LoRA as low-rank matrix adaptation baseline where δW = B A^T with B initialized to zero (Section 3.1.2); (3) Tensorize ViT weights into a 4D MHA tensor W^mha in R^{3l×d_model×h×d_h} by stacking Q/K/V across heads and blocks, and a 3D FFN tensor W^ffn in R^{9l×d_model×d_model} by stacking W^O/W^up/W^down across blocks (Section 3.2.1); (4) Apply CPD-based low-rank updates δW^mha (4-factor CPD) and δW^ffn (3-factor CPD) with learned scaling vectors λ^A, λ^B and hyperparameter α, merged into frozen weights at inference (Section 3.2.2); (5) Compute gradients of the loss with respect to each CPD factor matrix by replacing the differentiated factor with an identity matrix in the outer product, leveraging PyTorch autograd (Section 3.3); (6) Initialize factor matrices: random normal for first factors A^(1)/B^(1), zeros for second factors A^(2)/B^(2) (ensuring δW=0 at start), orthogonal for remaining factors A^(3)/A^(4)/B^(3), and λ sampled from N(λ_μ, λ_σ²) (Section 3.4).",
      "source": "Section 3.1-3.4"
    }
  ]
}