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\title[Dual DeltaNet]{Dual DeltaNet}
\subtitle{Value-Side Memory Reorganization for Linear Attention}
\author{\texorpdfstring{Paper by Yixiao Qian, Yaxin Zhang, Song Chen,\\Jiaxu Liu, Shengze Cai, and Chao Xu}{Paper by Yixiao Qian, Yaxin Zhang, Song Chen, Jiaxu Liu, Shengze Cai, and Chao Xu}}
\date{arXiv:2610.38578v1 $\mid$ Research group reading $\mid$ 13 November 2026}
\begin{document}

\begin{frame}
\titlepage
\vspace{-6mm}
\begin{center}\small Reading presentation prepared with Codex; not an author presentation.\end{center}
\end{frame}

\begin{frame}{A second read direction improves fixed-state recall}
\begin{block}{Central idea}
Correct the value associated with a key, then correct the address response associated with selected value features.
\end{block}
\begin{itemize}
\item One matrix state; complementary key-side and value-side delta rules.
\item At 1B: NIAH average \alert{74.52\%}, versus KDA's 64.27\%.
\item Higher latency than KDA; gains vary with task and scale.
\end{itemize}
\medskip
{\small \href{https://arxiv.org/abs/2610.38578v1}{Paper: arXiv v1, submitted 28 October 2026}.\\
Publication not found as of 13 November 2026.\\
\href{https://github.com/ReichtumQian/DualDeltaNet}{Official code: DualDeltaNet} --- implementation and model configurations available.}
\papersource{Paper, Secs. 3--5, PDF pp. 3--8; Table 3. Publication/code checked 2026-11-13.}
\end{frame}

\begin{frame}{A fixed-size state forces associations to share space}
\[
S_t\in\mathbb R^{d_k\times d_v},\qquad
S_t=S_{t-1}+k_tv_t^\top,\qquad o_t=S_t^\top q_t
\]
\begin{itemize}
\item $q_t,k_t\in\mathbb R^{d_k}$; $v_t,o_t\in\mathbb R^{d_v}$.
\item Rows act as $d_k$ addresses storing $d_v$-dimensional content.
\item Fixed recurrent memory avoids a growing token-level KV cache.
\item Overlapping writes can interfere: useful associations must survive compression.
\end{itemize}
\begin{block}{Research question}
Can we improve memory organization while keeping the same state capacity?
\end{block}
\papersource{Paper, Secs. 1--2, PDF pp. 1--3, Eqs. (1)--(2).}
\end{frame}

\begin{frame}{KDA corrects the value retrieved by the current key}
\[
\bar S_{t-1}=D_tS_{t-1},\qquad D_t=\operatorname{Diag}(\alpha_t)
\]
\[
S_t^{\mathrm{KDA}}=\bar S_{t-1}
+\beta_t k_t\underbrace{(v_t-\bar S_{t-1}^\top k_t)^\top}_{\text{value residual}}
\]
\begin{itemize}
\item $q_t,k_t$ are L2-normalized; $\alpha_t\in(0,1)^{d_k}$.
\item Decay controls retention separately for each key channel.
\item $\beta_t\in(0,1)$ scales correction at the current key direction.
\item This read does not explicitly inspect where related value features reside.
\end{itemize}
\papersource{Paper, Secs. 2--3, PDF p. 3, Eqs. (4)--(6).}
\end{frame}

\begin{frame}{Reading from the value side reveals address occupancy}
\[
S^\top k=\sum_jv_j(k_j^\top k),\qquad
Se=\sum_jk_j(v_j^\top e)
\]
\begin{center}
\includegraphics[width=.94\linewidth,height=.40\textheight,keepaspectratio]{figures/fig-01-mechanism.pdf}
\end{center}
\begin{itemize}
\item $e\in\mathbb R^{d_v}$ selects value features; $Se\in\mathbb R^{d_k}$ aggregates their addresses.
\item An address response need not recover an individual historical semantic key.
\end{itemize}
\papersource{Paper, Sec. 1, PDF p. 2; original Fig. 1, PDF p. 3. Discrete relocation is schematic.}
\end{frame}

\begin{frame}{Select features for reading; use the value direction for writing}
\[
\hat v_t=\frac{v_t}{\norm{v_t}},\quad m_t=\operatorname{sigmoid}(s_t^m),\quad
e_t=\frac{m_t\odot\hat v_t}{\norm{m_t\odot\hat v_t}},\quad \tau_t=v_t^\top e_t
\]
\begin{block}{Value-side delta rule}
\[
S_t=S_t^{\mathrm{KDA}}+
\gamma_t\underbrace{(\tau_tk_t-S_t^{\mathrm{KDA}}e_t)}_{\text{address residual}}\hat v_t^\top
\]
\end{block}
\begin{itemize}
\item $\gamma_t=\operatorname{sigmoid}(s_t^\gamma)$ controls correction strength.
\item $\tau_tk_t$ is the read response of the target association $k_tv_t^\top$.
\item The selector $e_t$ probes memory; $\hat v_t$ is the write direction.
\end{itemize}
\papersource{Paper, Sec. 3, PDF p. 3, Eqs. (7)--(8).}
\end{frame}

\begin{frame}{The selected address response moves toward its target}
Let $r=S_t^{\mathrm{KDA}}e_t$ and $c_t=\hat v_t^\top e_t$. Then
\[
S_te_t=(1-\gamma_tc_t)r+\gamma_tc_t\tau_tk_t.
\]
\begin{itemize}
\item Positive channel gates imply $0<c_t\leq1$ for a nonzero value.
\item The selected response interpolates toward the target address response.
\item With $e_t=\hat v_t$, $c_t=1$: the effective step is exactly $\gamma_t$.
\item A continuous rank-one correction changes rows of the shared matrix.
\end{itemize}
\begin{block}{Interpretation}
The rule acts on an aggregate response; other stored associations can also change.
\end{block}
\papersource{Teaching derivation from paper Eq. (8), PDF p. 3; not a separate theorem in the paper.}
\end{frame}

\begin{frame}{Left and right transitions act on the same memory}
\[
A_t=I-\beta_tk_tk_t^\top,\qquad
B_t=I-\gamma_te_t\hat v_t^\top
\]
\begin{block}{Equivalent two-sided recurrence}
\[
S_t=A_tD_tS_{t-1}B_t+\eta_tk_t\hat v_t^\top
\]
\[
\eta_t=\beta_t\norm{v_t}+\gamma_t(1-\beta_t)\tau_t
\]
\end{block}
\begin{itemize}
\item $A_t$: key-side erasure; $B_t$: value-feature transition, generally asymmetric.
\item The effective write strength includes both correction stages.
\item Setting $\gamma_t=0$ recovers KDA.
\end{itemize}
\papersource{Paper, Sec. 3, PDF p. 3, Eqs. (9)--(10).}
\end{frame}

\begin{frame}{A rank-2 increment fuses both corrections for decoding}
Read both sides from the decayed state:
\[
\ell_t=\bar S_{t-1}^\top k_t,\quad r_t=\bar S_{t-1}e_t,\quad
\tilde v_t=v_t-\ell_t
\]
\[
\tilde k_t=(\tau_t-\beta_t\tilde v_t^\top e_t)k_t-r_t
\]
Pack two outer products:
\[
\mathcal K_t=[k_t\ \tilde k_t],\quad
\mathcal V_t=\begin{bmatrix}\beta_t\tilde v_t^\top\\\gamma_t\hat v_t^\top\end{bmatrix},\quad
S_t=\bar S_{t-1}+\mathcal K_t\mathcal V_t
\]
\begin{itemize}
\item The correction in $\tilde k_t$ accounts for the intervening KDA update.
\item Increment rank $\leq2$; the full state can have much higher rank.
\item Readout: $o_t=d_k^{-1/2}S_t^\top q_t$.
\end{itemize}
\papersource{Paper, Algorithm 1, PDF p. 4.}
\end{frame}

\begin{frame}{More computation shares the same leading state traffic}
\begin{center}
\begin{tabular}{lcc}
\toprule & KDA & DDN \\
\midrule
Leading FLOPs/token & $7d_kd_v$ & $11d_kd_v$ \\
State load + store & $4d_kd_v$ bytes & $4d_kd_v$ bytes \\
Asymptotic intensity & 1.75 FLOPs/byte & 2.75 FLOPs/byte \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Assumption: 2-byte state/input/output elements, state traffic dominant.
\item Additional value-side input is only $\mathcal O(d_v)$.
\item Theoretical intensity rises by 57.1\%; this can improve memory-bound utilization.
\item Actual latency also depends on reductions, scheduling, and kernel parallelism.
\end{itemize}
\papersource{Paper, Table 1 and Sec. 3, PDF p. 4. Percent change calculated from 2.75/1.75.}
\end{frame}

\begin{frame}{Chunkwise training first removes channel-wise decay}
Within a length-$C$ chunk, suppress the chunk index and define
\[
\Pi^i=\operatorname{Diag}(\alpha^i\odot\cdots\odot\alpha^1),\qquad
\widehat S^i=(\Pi^i)^{-1}S^i.
\]
\[
\bar k^i=(\Pi^i)^{-1}k^i,\quad \bar u^i=\beta^i\Pi^ik^i,\quad
\bar A^i=I-\bar k^i(\bar u^i)^\top
\]
\[
\widehat S^i=\bar A^i\widehat S^{i-1}B^i+
\eta^i\bar k^i(\hat v^i)^\top
\]
\begin{itemize}
\item Unrolling gives $P_LS_{\mathrm{in}}P_R+\sum_i\eta^ih^i(z^i)^\top$.
\item Left and right products propagate each write to the chunk boundary.
\item All $\alpha$ are positive; implementation uses FP32 cumulative log-decay.
\end{itemize}
\papersource{Paper, Sec. 4, PDF pp. 4--5, Eqs. (11)--(16); App. B, PDF p. 11.}
\end{frame}

\begin{frame}{Two compact WY factors replace long transition products}
Stack $\bar k^i,\bar u^i,e^i,\hat v^i$ as rows of $\bar K,\bar U,E,\hat V$.
\[
T_L=\tril(\bar U\bar K^\top,-1),\quad
T_R=\tril(\Gamma E\hat V^\top,-1),\quad \Gamma=\operatorname{Diag}(\gamma^i)
\]
\[
M_L=(I+T_L)^{-1},\quad M_R=(I+T_R)^{-1},\quad
H=\bar K^\top M_L,\quad Z=M_R^\top\hat V
\]
\begin{block}{Chunk boundary, with $N=\operatorname{Diag}(\eta^i)$}
\[
\widehat S^C=(I-H\bar U)S_{\mathrm{in}}(I-E^\top\Gamma Z)+HNZ
\]
\end{block}
\begin{itemize}
\item $H$: surviving key directions; $Z$: surviving value directions.
\item Unit-triangular systems encode within-chunk interactions.
\item Chunks still exchange recurrent boundary states sequentially.
\end{itemize}
\papersource{Paper, Sec. 4, PDF p. 5, Eqs. (17)--(25); Lemma A.1, PDF p. 11.}
\end{frame}

\begin{frame}{Causal masks keep every query inside its own prefix}
\[
(Q_\pi)_{j,:}=d_k^{-1/2}(\pi^j\odot q^j)^\top,\qquad F_R=M_R\Gamma E
\]
\[
\begin{aligned}
G_L&=\tril(Q_\pi\bar K^\top)M_L,\\
Q_{\mathrm{eff}}&=Q_\pi-G_L\bar U,\qquad
O_{\mathrm{base}}=Q_{\mathrm{eff}}S_{\mathrm{in}},\\
\Phi&=G_LNM_R^\top,\\
O&=O_{\mathrm{base}}+\tril(\Phi-O_{\mathrm{base}}F_R^\top)\hat V.
\end{aligned}
\]
\begin{itemize}
\item $O_{\mathrm{base}}$: incoming-state contribution after left-side transitions.
\item The remaining term combines writes and right-side correction.
\item Here $\tril$ includes the diagonal: token $j$ may read its own update.
\end{itemize}
\papersource{Paper, Sec. 4, PDF pp. 5--6, Eqs. (26)--(33), Algorithm 2.}
\end{frame}

\begin{frame}{Value-side reductions constrain kernel parallelism}
\begin{columns}[T]
\begin{column}{.39\textwidth}
\centering
\includegraphics[width=.97\linewidth,height=.67\textheight,keepaspectratio]{figures/fig-02-block.pdf}
\end{column}
\begin{column}{.59\textwidth}
\begin{itemize}
\item KDA-like projections, causal convolutions, SiLU; normalized $q,k$.
\item New gates: feature selector $m_t$ and correction strength $\gamma_t$.
\item The $Se$ reduction couples value columns, limiting independent value tiling.
\item TileLang kernel: one program per sequence/head; full state in shared memory.
\item Small batch sizes can leave too few independent programs.
\end{itemize}
\end{column}
\end{columns}
\papersource{Original Fig. 2, PDF p. 4; App. B, PDF p. 11.}
\end{frame}

\begin{frame}{Comparisons match tokens, total parameters, and state size}
\begin{center}
\begin{tabular}{lrrr}
\toprule Scale & Tokens & Layers & State elements/layer \\
\midrule
130M & $\sim$5B & 12 & 98,304 \\
340M & $\sim$15B & 24 & 131,072 \\
1B & $\sim$50B & 24 & 229,376 \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item FineWeb-Edu; 2,048-token training context; batch 256 sequences.
\item Shared LLaMA tokenizer (32K vocabulary), AdamW and cosine schedule.
\item Baselines: KDA, GDN-2, Mamba-3 SISO and rank-4 MIMO.
\item Model/MLP widths are adjusted: equal totals do not imply identical blocks.
\end{itemize}
\papersource{Paper, Sec. 5, PDF p. 6; App. C, Tables 7--8, PDF pp. 11--12.}
\end{frame}

\begin{frame}{General modeling improves modestly over KDA}
\begin{center}
\begin{tabular}{llrrrr}
\toprule Scale & Model & Train ppl $\downarrow$ & WT2 ppl $\downarrow$ & LAMB. ppl $\downarrow$ & Avg.\ acc.\ $\uparrow$ \\
\midrule
130M & KDA & 14.79 & 42.53 & 69.91 & 37.38 \\
     & DDN & 14.78 & 42.36 & 60.93 & \textbf{38.03} \\
340M & KDA & 10.93 & 28.13 & 29.09 & 42.40 \\
     & DDN & 10.88 & 27.43 & 28.41 & \textbf{43.44} \\
1B   & KDA & 8.23 & 19.07 & 13.86 & 50.37 \\
     & DDN & 8.22 & 19.00 & 13.56 & \textbf{50.59} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Seven-task mean is highest among all tested models at each scale.
\item At 1B, DDN trails KDA on WinoGrande: 54.46\% vs.\ 56.12\%.
\item Training-ppl gaps can be 0.01; run-to-run variation is not reported.
\end{itemize}
\papersource{Paper, Table 2, PDF p. 7. Accuracies are percentages; HellaSwag uses normalized accuracy.}
\end{frame}

\begin{frame}{NIAH provides the strongest average gains}
\begin{center}
\begin{tabular}{lrrrrr}
\toprule Scale & M3 SISO & M3 MIMO & GDN-2 & KDA & DDN \\
\midrule
130M & 12.95 & 11.65 & 26.88 & 38.10 & \textbf{40.57} \\
340M & 35.02 & 41.53 & 48.57 & 51.20 & \textbf{53.87} \\
1B & 47.25 & 60.73 & 69.40 & 64.27 & \textbf{74.52} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Mean accuracy (\%) over four RULER variants and three context lengths.
\item Contexts: 1,024 / 2,048 / 4,096; 500 examples per variant/length.
\item Tasks: repeated-text numbers, essay numbers, essay UUIDs, multiple keys.
\item At 1B, DDN improves over KDA by \alert{10.25 percentage points}.
\end{itemize}
\papersource{Paper, Table 3, PDF p. 7; evaluation definitions in App. C, PDF p. 12.}
\end{frame}

\begin{frame}{The average hides a major 340M Single-2 regression}
\begin{center}
\begin{tabular}{llrrr}
\toprule Scale / task & Context & KDA (\%) & DDN (\%) & Difference (pp) \\
\midrule
1B / Single-2 & 4,096 & 54.2 & 80.8 & +26.6 \\
1B / Single-3 & 1,024 & 59.4 & 97.0 & +37.6 \\
1B / Single-3 & 4,096 & 19.0 & 36.0 & +17.0 \\
\midrule
340M / Single-2 & 2,048 & 95.6 & 67.2 & \alert{$-28.4$} \\
340M / Single-2 & 4,096 & 40.6 & 14.4 & \alert{$-26.2$} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item At 340M, Single-1/Single-3 improvements outweigh Single-2 losses.
\item MultiKey-1 also remains difficult; at 1B/1K, M3 MIMO scores 53.2\%, DDN 37.6\%.
\item The paper does not explain the Single-2 regression.
\end{itemize}
\papersource{Paper, Table 3, PDF p. 7. Selected cells; differences calculated from original values.}
\end{frame}

\begin{frame}{Real-world retrieval favors different updates by task}
\begin{center}
\begin{tabular}{lrrr}
\toprule Scale & KDA & GDN-2 & DDN \\
\midrule
130M & 15.18 & 15.53 & \textbf{16.18} \\
340M & 21.31 & 21.54 & \textbf{23.22} \\
1B & 30.90 & \textbf{31.80} & 31.20 \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Six-task mean contains accuracy (\%); inputs capped at 2,048 tokens.
\item SWDE, SQuAD Completion, FDA, TriviaQA/NQ/DROP Cloze.
\item At 1B, GDN-2's FDA score is 17.42\%, versus DDN's 12.89\%.
\item DDN leads NQ (17.45\%), but trails KDA on SWDE (32.58\% vs.\ 35.46\%).
\end{itemize}
\papersource{Paper, Table 4, PDF p. 8; App. C, PDF p. 12. Greedy generation, at most 48 new tokens.}
\end{frame}

\begin{frame}{Training speed depends on memory strategy and parallelism}
\begin{center}
\includegraphics[width=.48\linewidth,height=.58\textheight,keepaspectratio]{figures/fig-03a-throughput.pdf}\hfill
\includegraphics[width=.48\linewidth,height=.58\textheight,keepaspectratio]{figures/fig-03b-memory.pdf}
\par\smallskip
{\small
\textcolor[RGB]{69,122,179}{M3 SISO}\quad
\textcolor[RGB]{255,151,62}{M3 MIMO}\quad
\textcolor[RGB]{45,145,58}{GDN-2}\quad
\textcolor[RGB]{172,101,185}{KDA}\quad
\textcolor[RGB]{225,38,52}{DDN}}
\end{center}
\begin{itemize}
\item 1B, BF16, H200; roughly fixed tokens. DDN slows at 8K $\times$ 4.
\item Default recomputation differs: SISO spends more memory for speed.
\end{itemize}
\papersource{Original Fig. 3 panels, PDF p. 8; legend retypeset. See App. B, PDF p. 11 for parallelism.}
\end{frame}

\begin{frame}{DDN adds about 14\% total inference latency over KDA}
\begin{center}
\begin{tabular}{lrrrr}
\toprule & \multicolumn{2}{c}{4,096 tokens} & \multicolumn{2}{c}{8,192 tokens} \\
Model & Prefill & Prefill+Decode & Prefill & Prefill+Decode \\
\midrule
M3 SISO & 1.35 & 14.66 & 2.69 & 27.44 \\
M3 MIMO & 1.64 & 16.70 & 3.35 & 33.46 \\
GDN-2 & 1.58 & 16.53 & 3.16 & 33.11 \\
KDA & 1.39 & 14.66 & 2.81 & 29.34 \\
DDN & 1.57 & 16.68 & 3.15 & 33.37 \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Seconds; 1B, H200, batch 64; generated length equals input length.
\item At 4K: prefill +12.95\%; prefill+decode +13.78\% versus KDA.
\item DDN latency is close to GDN-2 and M3 MIMO.
\end{itemize}
\papersource{Paper, Table 5, PDF p. 8. Overheads calculated as (DDN/KDA $-1$)$\times$100\%.}
\end{frame}

\begin{frame}{Feature gating helps NIAH, with a retrieval tradeoff}
\begin{center}
\begin{tabular}{llrrr}
\toprule Scale & Variant & Downstream & Retrieval & NIAH \\
\midrule
130M & KDA & 37.38 & 15.18 & 38.10 \\
 & DDN, $e=\hat v$ & 37.41 & 15.87 & 39.45 \\
 & DDN, gated & \textbf{38.03} & \textbf{16.18} & \textbf{40.57} \\
340M & KDA & 42.40 & 21.30 & 51.20 \\
 & DDN, $e=\hat v$ & 42.16 & \textbf{23.48} & 52.42 \\
 & DDN, gated & \textbf{43.44} & 23.22 & \textbf{53.87} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item All columns are mean accuracies (\%). Both DDN variants keep the value-side rule.
\item Gating improves NIAH by 1.12/1.45 pp over ungated DDN.
\item 340M retrieval favors the ungated selector by 0.26 pp.
\end{itemize}
\papersource{Paper, Table 6, PDF p. 9. Its 340M KDA retrieval is 21.30; Table 4 reports 21.31.}
\end{frame}

\begin{frame}{The next controls should explain when reorganization helps}
\begin{itemize}
\item \textbf{Attribution:} a parameter-matched $\gamma=0$ control would help isolate the value-side correction from architecture adjustments.
\item \textbf{Failure mechanism:} why does 340M Single-2 regress by 26--28 pp while other variants improve?
\item \textbf{Precision of small gains:} repeated training would clarify the 0.01-ppl and sub-point downstream differences.
\item \textbf{Deployment:} value-column coupling limits small-batch parallelism; the current benchmark uses batch 64.
\end{itemize}
\begin{block}{Reading assessment}
Strong algebraic specification and promising recall results; task-specific behavior and kernel cost remain central to adoption.
\end{block}
\papersource{Analysis based on Tables 2--6, PDF pp. 7--9; architecture/implementation, App. B--C, pp. 11--12.}
\end{frame}

\begin{frame}{Two-sided memory maintenance is the main contribution}
\begin{enumerate}
\item \textbf{Mechanism:} key-side value correction and value-side address-response correction share one fixed-size state.
\medskip
\item \textbf{Computation:} two-sided compact WY supports chunkwise training; a rank-2 increment fuses recurrent decoding.
\medskip
\item \textbf{Evidence:} the largest gains appear in NIAH; real retrieval and latency expose the practical tradeoffs.
\end{enumerate}
\papersource{Paper, Secs. 3--6, PDF pp. 3--9. Main talk: approximately 25 minutes; derivation/details follow.}
\end{frame}

\begin{frame}{References and implementation}
\begin{itemize}
\item Qian et al.\ (2026). \emph{Dual DeltaNet: Value-Side Memory Reorganization for Linear Attention}.\\
\href{https://arxiv.org/abs/2610.38578v1}{arXiv:2610.38578v1}; all paper locators refer to this version.
\medskip
\item \href{https://github.com/ReichtumQian/DualDeltaNet}{Official implementation: ReichtumQian/DualDeltaNet}.\\
Includes DDN layers, models, kernels and training configurations.
\medskip
\item \href{https://arxiv.org/abs/2610.38578}{Current arXiv identity page}: no formal publication record found as of 13 November 2026.
\end{itemize}
\papersource{Figures 1--2 cropped from the paper; Fig. 3 reused from author TeX assets. Tables are selected/retypeset original values.}
\end{frame}

\pdfstringdefDisableCommands{\def\translate#1{#1}}
\appendix
\begin{frame}{Backup: the compact WY identity follows by telescoping}
For $Q_i=I-u_iw_i^\top$, let $f_i=Q_m\cdots Q_{i+1}u_i$.
\[
Q_m\cdots Q_{i+1}=I-\sum_{j>i}f_jw_j^\top
\]
\[
f_i=u_i-\sum_{j>i}f_j(w_j^\top u_i)
\]
Stack columns in $F,U,W$ and set $T=\tril(W^\top U,-1)$:
\[
F(I+T)=U,\qquad Q_m\cdots Q_1=I-U(I+T)^{-1}W^\top.
\]
\begin{itemize}
\item $I+T$ is unit lower triangular, so it is always invertible.
\item No orthogonality assumption on $u_i,w_i$ is required.
\end{itemize}
\papersource{Paper, App. A, Lemma A.1, PDF p. 11, Eqs. (34)--(37).}
\end{frame}

\begin{frame}{Backup: collect the chunk boundary into matrix products}
With $H=\bar K^\top M_L$ and $Z=M_R^\top\hat V$,
\[
\begin{aligned}
F_R&=M_R\Gamma E,\\
R&=S_{\mathrm{in}}-S_{\mathrm{in}}F_R^\top\hat V,\\
\Psi&=NZ-\bar U R,\\
S_{\mathrm{out}}&=\Pi^C(R+H\Psi).
\end{aligned}
\]
\begin{itemize}
\item $R$: incoming state after all right-side transitions in the chunk.
\item $H\Psi$: collects new writes and left-side correction.
\item Restore cumulative decay only after the normalized boundary update.
\end{itemize}
\papersource{Paper, Sec. 4, PDF p. 5, Eqs. (23)--(25), Algorithm 2.}
\end{frame}

\begin{frame}{Backup: parameterization, precision, and training details}
\[
\alpha_t=\exp[-\exp(a_{\mathrm{decay}})\odot\Delta_t],\qquad
\Delta_t=\operatorname{softplus}(W_\Delta x_t+b_\Delta)
\]
\begin{itemize}
\item Separate low-rank decay/selector projections; bottleneck width $d_v$.
\item Initialize $m_t=0.5$, $\gamma_t=0.05$, hence $e_t=\hat v_t$ initially.
\item BF16 operands/state; FP32 accumulation, normalization, log-decay and triangular intermediates.
\item AdamW: $(0.9,0.95)$, $\epsilon=10^{-8}$, decay 0.01, gradient clip 1.0.
\item Steps: 9,500 / 30,000 / 95,000; peak LR: $6\!\times\!10^{-4}$ / $4\!\times\!10^{-4}$ / $4\!\times\!10^{-4}$.
\item NIAH: greedy decoding, at most 128 new tokens; real retrieval: 48.
\end{itemize}
\papersource{Paper, App. B--C, PDF pp. 11--12, Eqs. (38)--(39), Tables 7--8.}
\end{frame}
\end{document}