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fla/ops/delta_rule/README.md (5)

5-11: Fix inline math fencing and tighten notation.

• The inline equation uses mixed backticks inside math: $…$. This will render incorrectly in many Markdown engines.
• Minor: vectors/matrices alternate between \bf{} and \mathbf{}; prefer a single convention (typically \mathbf{}) for consistency.

Apply this diff to fix the inline math and unify boldface for the shown line:

-To reduce notational clutter, we focus on the first chunk, denoting $\mathbf{S}^r=\mathbf{S}_{[1]}^r$. By unrolling the recurrence $`S_t = S_{t-1}(I - \beta_t k_t k_t^\top) + \beta_t v_t k_t^\top`$, we have:
+To reduce notational clutter, we focus on the first chunk, denoting $\mathbf{S}^r=\mathbf{S}_{[1]}^r$. By unrolling the recurrence $S_t = S_{t-1}(\mathbf{I} - \beta_t \mathbf{k}_t \mathbf{k}_t^\top) + \beta_t \mathbf{v}_t \mathbf{k}_t^\top$, we have:

Optionally add a short “Notation” bullet list (dims for $\mathbf{K},\mathbf{V},\mathbf{Q},\mathbf{S}^0$, and chunk length $C$) right after this paragraph.

---

20-22: WY form looks right; keep symbol style consistent.

Derivation and dims check out. Consider switching \bf{} → \mathbf{} for $,\mathbf{w}^i,\mathbf{k}^i,$ to match the rest.

---

56-62: Matrix forms for P and H are dimensionally consistent.

$(\mathbf{W}^\top\mathbf{K})=\sum_i \mathbf{w}^i\mathbf{k}^{i\top}$ and $(\mathbf{U}^\top\mathbf{K})=\sum_i \mathbf{u}^i\mathbf{k}^{i\top}$ are accurate. Consider stating shapes of $\mathbf{K}\in\mathbb{R}^{C\times d_k}$ and $\mathbf{V}\in\mathbb{R}^{C\times d_v}$ explicitly here.

---

65-71: Avoid explicit matrix inverse in implementations.

The triangular solve form is correct, but writing $(\cdot)^{-1}$ can invite naïve implementations. Recommend phrasing as “solve lower‑triangular system” to encourage using forward substitution instead of forming the inverse.

Proposed wording tweak:

-\mathbf{W} &= \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\mathrm{diag}(\beta) \mathbf{K}\\
-\mathbf{U} &= \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\mathrm{diag}(\beta) \mathbf{V}
+\text{Solve } \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta)\,\mathbf{K}\mathbf{K}^\top,-1)\right)\mathbf{W}=\mathrm{diag}(\beta)\,\mathbf{K}\\
+\text{Solve } \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta)\,\mathbf{K}\mathbf{K}^\top,-1)\right)\mathbf{U}=\mathrm{diag}(\beta)\,\mathbf{V}

---

76-81: Use boldface for T consistently and define “UT transform.”

• Lines 79–80 use T instead of \mathbf{T}.
• “UT transform” is introduced but not expanded; clarify it (e.g., unit‑lower triangular transform scaled by diag(β)).

Apply this diff:

-\mathbf{P} &= \mathbf{I} - \mathbf{K}^\top T^\top \mathbf{K} \;\;\in \mathbb{R}^{d_k \times d_k}\\
-\mathbf{H} &= \mathbf{V}^\top T^\top \mathbf{K} \;\;\in \mathbb{R}^{d_v \times d_k}
+\mathbf{P} &= \mathbf{I} - \mathbf{K}^\top \mathbf{T}^\top \mathbf{K} \;\;\in \mathbb{R}^{d_k \times d_k}\\
+\mathbf{H} &= \mathbf{V}^\top \mathbf{T}^\top \mathbf{K} \;\;\in \mathbb{R}^{d_v \times d_k}

And add a parenthetical after the first mention of $\mathbf{T}$: “(a unit‑lower triangular transform scaled by $\mathrm{diag}(\beta)$)”.

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fla/ops/delta_rule/README.md (2)

5-5: Remove backticks inside inline math.

Backticks render code, not math. Use plain LaTeX inside $…$.

-To reduce notational clutter, we focus on the first chunk, denoting $\mathbf{S}^r=\mathbf{S}_{[1]}^r$. By unrolling the recurrence $`S_t = S_{t-1}(I - \beta_t k_t k_t^\top) + \beta_t v_t k_t^\top`$, we have:
+To reduce notational clutter, we focus on the first chunk, denoting $\mathbf{S}^r=\mathbf{S}_{[1]}^r$. By unrolling the recurrence $S_t = S_{t-1}(I - \beta_t k_t k_t^\top) + \beta_t v_t k_t^\top$, we have:

---

91-92: Define mask M right after the equations.

You use $\mathbf{M}$ but don’t define it; add a one‑liner.

 \end{equation}

+\noindent$\mathbf{M}\in\mathbb{R}^{C\times C}$ denotes the causal lower‑triangular (0/1) mask induced by the UT structure (ones on allowed causal positions, zeros elsewhere). The Hadamard product $\odot$ enforces causal masking of $\mathbf{Q}_{[t]}\mathbf{K}^\top$, matching the strictly lower‑triangular support of $\mathbf{T}$.
+
In this final form, the intra-chunk recurrence has been transformed into a series of efficient matrix multiplications (e.g., computing $\mathbf{T}$, $\mathbf{W}$, $\mathbf{U}$, and the final output), which can be highly optimized on modern hardware like GPUs.

</blockquote></details>

</blockquote></details>

<details>
<summary>🧹 Nitpick comments (4)</summary><blockquote>

<details>
<summary>fla/ops/delta_rule/README.md (4)</summary><blockquote>

`9-11`: **Use \mathbf consistently instead of deprecated \bf.**

Standardize bold symbols for vectors/matrices.


```diff
-... (\mathbf{I} - \beta^i \bf{k}^i \bf{k}^{i\top}) ...
-... \beta^i \bf{v}^i\bf{k}^{i\top} ...
+... (\mathbf{I} - \beta^i \mathbf{k}^i \mathbf{k}^{i\top}) ...
+... \beta^i \mathbf{v}^i\mathbf{k}^{i\top} ...

-\mathbf{P}^{r} = \mathbf{I} - \sum_{i=1}^{r}\bf{w}^i\bf{k}^{i\top}
-\bf{w}^r = \beta^r \left(\bf{k}^r -  \sum_{i=1}^{r-1} \left(\bf{k}^{i\top}\bf{k}^r \right)\bf{w}^i  \right)
+\mathbf{P}^{r} = \mathbf{I} - \sum_{i=1}^{r}\mathbf{w}^i\mathbf{k}^{i\top}
+\mathbf{w}^r = \beta^r \left(\mathbf{k}^r -  \sum_{i=1}^{r-1} \left(\mathbf{k}^{i\top}\mathbf{k}^r \right)\mathbf{w}^i  \right)

-... \beta^r \bf{k}^r \bf{k}^{r\top} ...
-... \sum_{i=1}^{r-1}\bf{w}^i\bf{k}^{i\top} ...
+... \beta^r \mathbf{k}^r \mathbf{k}^{r\top} ...
+... \sum_{i=1}^{r-1}\mathbf{w}^i\mathbf{k}^{i\top} ...

-\mathbf{H}^{r} = \sum_{i=1}^{r} \bf{u}^i \bf{k}^{i\top}
-\bf{u}^r = \beta^r \left(\bf{v}^r -  \sum_{i=1}^{r-1} \left(\bf{k}^{i\top}\bf{k}^r\right) \bf{u}^i \right)
+\mathbf{H}^{r} = \sum_{i=1}^{r} \mathbf{u}^i \mathbf{k}^{i\top}
+\mathbf{u}^r = \beta^r \left(\mathbf{v}^r -  \sum_{i=1}^{r-1} \left(\mathbf{k}^{i\top}\mathbf{k}^r\right) \mathbf{u}^i \right)

-... \sum_{i=1}^{r-1}\bf{u}^i \bf{k}^{i\top} ...
-... \beta^r \bf{v}^r \bf{k}^{r\top} ...
+... \sum_{i=1}^{r-1}\mathbf{u}^i \mathbf{k}^{i\top} ...
+... \beta^r \mathbf{v}^r \mathbf{k}^{r\top} ...

Also applies to: 20-22, 28-34, 40-41, 47-53

---

68-70: Prefer \operatorname for operators (diag, tril).

Improves LaTeX typesetting; math operators won’t appear in roman text.

-\mathbf{W} &= \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\mathrm{diag}(\beta) \mathbf{K}\\
-\mathbf{U} &= \left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\mathrm{diag}(\beta) \mathbf{V}
+\mathbf{W} &= \left(\mathbf{I} + \operatorname{tril}(\operatorname{diag}(\beta)\,\mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\operatorname{diag}(\beta)\,\mathbf{K}\\
+\mathbf{U} &= \left(\mathbf{I} + \operatorname{tril}(\operatorname{diag}(\beta)\,\mathbf{K}\mathbf{K}^\top, -1)\right)^{-1}\operatorname{diag}(\beta)\,\mathbf{V}

-\mathbf{T} &= \left(\mathbf{I} + \mathrm{tril}\left(\mathrm{diag}(\beta)\mathbf{K} \mathbf{K}^\top,-1\right)\right)^{-1}\mathrm{diag}\left(\beta\right)
+\mathbf{T} &= \left(\mathbf{I} + \operatorname{tril}\!\left(\operatorname{diag}(\beta)\,\mathbf{K} \mathbf{K}^\top,-1\right)\right)^{-1}\operatorname{diag}\!\left(\beta\right)

Also applies to: 76-76

---

79-80: Boldface T for consistency.

Use $\mathbf{T}$ where introduced as a matrix.

-\mathbf{P} &= \mathbf{I} - \mathbf{K}^\top T^\top \mathbf{K} \;\;\in \mathbb{R}^{d_k \times d_k}\\
-\mathbf{H} &= \mathbf{V}^\top T^\top \mathbf{K} \;\;\in \mathbb{R}^{d_v \times d_k}
+\mathbf{P} &= \mathbf{I} - \mathbf{K}^\top \mathbf{T}^\top \mathbf{K} \;\;\in \mathbb{R}^{d_k \times d_k}\\
+\mathbf{H} &= \mathbf{V}^\top \mathbf{T}^\top \mathbf{K} \;\;\in \mathbb{R}^{d_v \times d_k}

---

84-85: Also remove backticks around bold symbols in this sentence.

Inline code formatting breaks math rendering.

-Substituting these compact forms back into the state update and output equations yields the hardware-efficient chunkwise algorithm. For a given chunk $[t]$ with initial state $`\mathbf{S}^0_{[t]}`$, the final state $`\mathbf{S}_{[t+1]}`$ and output $`\mathbf{O}_{[t]}`$ are:
+Substituting these compact forms back into the state update and output equations yields the hardware-efficient chunkwise algorithm. For a given chunk $[t]$ with initial state $\mathbf{S}^0_{[t]}$, the final state $\mathbf{S}_{[t+1]}$ and output $\mathbf{O}_{[t]}$ are:

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