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\title[Attention at any temperature]{Subquadratic Algorithms and Hardness\texorpdfstring{\\}{ }for Attention with Any Temperature}
\subtitle{\texorpdfstring{Paper reading and proof discussion\quad$\cdot$\quad 20--30 minutes}{Paper reading and proof discussion, 20--30 minutes}}
\author{Paper authors: Shreya Gupta, Boyang Huang, Barna Saha,\texorpdfstring{\\}{ }Yinzhan Xu, Christopher Ye}
\institute{\texorpdfstring{University of California, San Diego\quad$|$\quad Reader-prepared presentation}{UC San Diego; reader-prepared presentation}}
\date{\texorpdfstring{\href{https://arxiv.org/abs/2505.14840v1}{arXiv:2505.14840v1}\quad$\cdot$\quad Read on September 28, 2026}{arXiv:2505.14840v1; September 28, 2026}}

\begin{document}

% 1
\begin{frame}
  \titlepage
  \vspace{-4mm}
  \begin{center}
    \alert{Fixed dimension permits high-precision attention without a high-temperature restriction.}
  \end{center}
\end{frame}

% 2
\begin{frame}{The output is small; the guarantee is strong}
  \[
  Q,K,V\in[-B,B]^{n\times d},\qquad
  O=D^{-1}AV,\quad A=e^{QK^\top},\quad D=\diag(A\one).
  \]
  \begin{block}{Approximate Attention Computation}
    Return $\widehat O$ with
    $\displaystyle\|\widehat O-O\|_\infty
       :=\max_{i,t}|\widehat O_{i,t}-O_{i,t}|<\varepsilon$.
  \end{block}
  \begin{itemize}
    \item $n$: context length; $d$: head dimension; $B$: entry bound.
    \item Explicit pairwise computation costs $O(n^2d)$; output size is $nd$.
    \item The target is an error guarantee for \emph{every output entry}.
  \end{itemize}
  \papersource{Definitions 1.1--1.2, PDF p. 2. Exponentiation is entrywise.}
\end{frame}

% 3
\begin{frame}{Temperature becomes an input-magnitude parameter}
  \[
    e^{QK^\top/T}
    =e^{(Q/\sqrt T)(K/\sqrt T)^\top},\qquad T>0.
  \]
  \begin{itemize}
    \item A smaller temperature can create large, sharply separated scores.
    \item Seek runtime polynomial in the \alert{representation length} of the magnitude: $\polylog B$.
    \item With unchanged values, use $B_T=\max\{B,B/\sqrt T\}$.
  \end{itemize}
  \begin{block}{A comparison needs a common scale}
    This paper uses $e^{QK^\top}$; Alman--Song uses $e^{QK^\top/d}$.
    Entry-bound thresholds change under this rescaling.
  \end{block}
  \papersource{Introduction, PDF pp. 2--3; footnote 9, p. 17. External definition check: arXiv:2302.13214v2.}
\end{frame}

% 4
\begin{frame}{Dimension separates the main algorithmic regimes}
  \centering
  \begin{tabular}{@{}lll@{}}
    \toprule
    Dimension & Main bound & Conditions \\
    \midrule
    $d=O(1)$ & $\widetilde O(n^{2-1/d}\polylog(B/\varepsilon))$ & Algorithm \\[2mm]
    $d=2^{C\log^* n}$ & $n^{2-\delta}$ & SETH; suitable $C(\delta)$ \\[2mm]
    $d=\poly(n)$ & $\tmul(n,d,n)^{1-o(1)}$ & Generalized high-dim.
    \\
    & & OV hypothesis \\
    \bottomrule
  \end{tabular}
  \vspace{3mm}
  \begin{itemize}
    \item Small-dimension hardness: polynomial $B$, constant error.
    \item Large-dimension hardness: $B=O(\sqrt{\log n})$, constant error.
    \item Fixed-$d$ upper-bound exponents are \emph{not} proved optimal.
  \end{itemize}
  \papersource{Theorems 1.3, 5.4, 5.5; PDF pp. 3, 27. Adapted from Table 1, p. 5.}
\end{frame}

% 5
\begin{frame}{The algorithm makes exponential approximation local}
  \begin{enumerate}
    \item \textbf{Trim the tail.} Retain scores close to the row maximum.
    \item \textbf{Shift the row.} Relevant scores occupy a logarithmic interval.
    \item \textbf{Approximate locally.} Use a low-degree polynomial for $e^x$.
    \item \textbf{Aggregate moments.} Answer weighted geometric range queries.
  \end{enumerate}
  \begin{block}{Why all four steps matter}
    Scores can have huge absolute magnitude, yet the retained interval has
    width $O(\log(nB/\varepsilon))$.
  \end{block}
  \papersource{Technical overview, PDF pp. 5--7; formal algorithm, Sec. 3, pp. 9--17.}
\end{frame}

% 6
\begin{frame}{A score gap controls the total discarded mass}
  Let $m_i=\max_jQ_i\cdot K_j$ and let $\eta$ denote internal accuracy.
  \[
    R_i=\{j:Q_i\cdot K_j\geq m_i-\log(n/\eta)\}.
  \]
  \[
    j\notin R_i\ \Longrightarrow\
    \frac{p_{ij}}{p_{i,\max}}\leq\frac{\eta}{n}
    \ \Longrightarrow\ \sum_{j\notin R_i}p_{ij}\leq\eta.
  \]
  \begin{itemize}
    \item Renormalizing retained weights changes an output entry by at most $3\eta B$.
    \item Values may be signed; the guarantee is \alert{additive}.
    \item $R_i$ can contain all keys: fast aggregation is still necessary.
  \end{itemize}
  \papersource{Sec. 3.1, PDF pp. 10--11. The formal multidimensional algorithm uses a rounded row maximum.}
\end{frame}

% 7
\begin{frame}{In one dimension, relevance is an interval}
  \begin{center}
    \includegraphics[width=0.86\linewidth,height=0.42\textheight,keepaspectratio]{figures/fig-01.pdf}
  \end{center}
  \begin{itemize}
    \item Positive query: retain a suffix in key order; negative query: reverse it.
    \item Figure 1 illustrates an initial rounding scheme, with polynomial accuracy cost.
    \item The final algorithm replaces fine bins with polynomial moments.
  \end{itemize}
  \papersource{Original Figure 1, PDF p. 5; Secs. 1.1 and 3.1. Bin width 3 is illustrative, not a valid small-error setting.}
\end{frame}

% 8
\begin{frame}{After shifting, a logarithmic-degree polynomial suffices}
  \[
    c_i=m_i-\log(n/\eta),\qquad
    Q_i\cdot K_j-c_i\in[0,\log(n/\eta)]\quad(j\in R_i).
  \]
  Choose $P$ of degree $g=O(\log(n/\eta))$ satisfying
  \[
    |P(x)-e^x|\leq\eta e^x.
  \]
  \begin{block}{Normalized approximation}
    \[
    \widehat O_{i,t}=
    \frac{\sum_{j\in R_i}P(Q_i\cdot K_j-c_i)V_{j,t}}
         {\sum_{j\in R_i}P(Q_i\cdot K_j-c_i)}.
    \]
  \end{block}
  Shift invariance preserves the relevant softmax distribution.
  \papersource{Lemma 3.3 and Eq. (1), PDF p. 11; multidimensional interval, p. 14.}
\end{frame}

% 9
\begin{frame}{Sorted prefix moments give near-linear time for $d=1$}
  \[
    \sum_{j\in R_i}P(q_i k_j-c_i)v_j
    =\sum_{\ell=0}^{g}m_\ell\sum_{b=0}^{\ell}
      \binom{\ell}{b}q_i^b(-c_i)^{\ell-b}\phi_v(i,b),
  \]
  \[
    \phi_v(i,b)=\sum_{j\in R_i}k_j^b v_j.
  \]
  \begin{itemize}
    \item Sort keys once; store prefix sums of $k_j^b v_j$ and $k_j^b$.
    \item Binary search finds each query's interval endpoint.
    \item Include $b=0$; a zero query returns the mean value.
  \end{itemize}
  \begin{center}
    \alert{$\widetilde O(n\polylog(B/\varepsilon))$ total time}
  \end{center}
  \papersource{Lemmas 3.1, 3.4--3.5; Algorithm 1; PDF pp. 9--13.}
\end{frame}

% 10
\begin{frame}{In higher dimensions, relevance is a halfspace}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.45\textwidth}
      \includegraphics[width=\linewidth,height=0.55\textheight,keepaspectratio]{figures/fig-02.pdf}
    \end{column}
    \begin{column}{0.51\textwidth}
      \[
      R_i=\{K_j:Q_i\cdot K_j\geq c_i\}.
      \]
      \begin{itemize}
        \item Black points: keys.
        \item Arrow: query $(2,1)$.
        \item Blue region: retained keys.
        \item Query directions differ; one sorted order no longer suffices.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Original Figure 2, PDF p. 7. Weights and coordinate scales are omitted in the original schematic.}
\end{frame}

% 11
\begin{frame}{Weighted range queries evaluate the polynomial}
  For a multi-index $\alpha$, $|\alpha|\leq g$, preprocess weights
  \[
    w_{j,\alpha,t}=\left(\prod_{k=1}^{d}K_{j,k}^{\alpha_k}\right)V_{j,t},
    \qquad \phi_t(i,\alpha)=\sum_{K_j\in R_i}w_{j,\alpha,t}.
  \]
  \begin{itemize}
    \item Multinomial expansion separates query factors from key moments.
    \item Build an extra set of moments without $V$ for the denominator.
    \item Matou\v{s}ek's structure: $O(n\log n)$ preprocessing per weight set; $\widetilde O(n^{1-1/d})$ per query.
    \item At most $g^{O(d)}$ moment types when $d$ is fixed.
  \end{itemize}
  \papersource{Theorem 3.7 and Lemmas 3.11--3.12, PDF pp. 13--16.}
\end{frame}

% 12
\begin{frame}{The error budget yields the main runtime bound}
  \[
    \underbrace{3\eta B}_{\text{discarded tail}}
    +\underbrace{4\eta B}_{\text{polynomial + normalization}}
    \leq\varepsilon
    \quad\text{with}\quad\eta\leq\frac{\varepsilon}{7B}.
  \]
  \[
    \widetilde O\!\left(
      n\,g^{O(d)}n^{1-1/d}\log(dB/\eta)\right)
    =\widetilde O\!\left(n^{2-1/d}\polylog(B/\varepsilon)\right).
  \]
  \begin{center}
    \begin{tabular}{@{}ccccc@{}}
      \toprule
      Fixed $d$ & 1 & 2 & 4 & 8 \\
      Exponent of $n$ & 1 & $3/2$ & $7/4$ & $15/8$ \\
      \bottomrule
    \end{tabular}
  \end{center}
  The hidden $g^{O(d)}$ dependence matters for practical dimensions.
  \papersource{Eq. (1), p. 11; Lemmas 3.10, 3.13, pp. 14, 16. Exponent table is a reader calculation, not a benchmark.}
\end{frame}

% 13
\begin{frame}{Low rank reduces score geometry; output cost needs care}
  If $Q=U_QV_Q^\top$ has rank $r$, then
  \[
    Q'=U_Q\in\R^{n\times r},\qquad
    K'=KV_Q\in\R^{n\times r},\qquad Q'K'^\top=QK^\top.
  \]
  \begin{block}{Theorem 1.4: reported bound}
    For $r=\min(\operatorname{rank}Q,\operatorname{rank}K)=O(1)$:
    $\widetilde O(nd+n^{2-1/r}\polylog(B/\varepsilon))$.
  \end{block}
  \begin{itemize}
    \item Exact score reduction is valid; $V$ still has $d$ output columns.
    \item The printed factorization has incompatible matrix dimensions.
    \item The proof does not account for removing the per-column query cost.
  \end{itemize}
  \papersource{Theorem 1.4 and proof, PDF pp. 16--17. Corrected score identity and qualifications: reader analysis.}
\end{frame}

% 14
\begin{frame}{The training result concerns one attention objective}
  \[
    H=A_3Y,\quad F=\softmax(A_1XA_2^\top),\quad
    Z=FH,\quad C=Z-E,
  \]
  \[
    L(X)=\tfrac12\|C\|_F^2,\qquad \text{target: }\nabla_XL.
  \]
  \begin{itemize}
    \item Theorem 4.3 reports $O(d)$ attention calls plus $O(nd^2)$ work.
    \item Reported oracle accuracy: $\varepsilon/\Theta(ndB^3)$.
    \item The printed gradient identity is inconsistent with row softmax.
  \end{itemize}
  \begin{block}{Scope}
    A single unit's gradient complexity does not establish end-to-end
    optimization cost or convergence for a full Transformer.
  \end{block}
  \papersource{Definitions 4.1--4.2, Theorem 4.3, PDF pp. 17--19; full Transformer left open on p. 30.}
\end{frame}

% 15
\begin{frame}{A corrected identity preserves the few-call structure}
  \textbf{Reader derivation:} let $r_i=\langle C_i,Z_i\rangle$ and compute
  \[
    U=FA_2,\qquad T_p=F\diag(H_{*,p})A_2.
  \]
  \[
    \nabla_XL=A_1^\top\left[
       \sum_{p=1}^{d}\diag(C_{*,p})T_p-\diag(r)U\right].
  \]
  \begin{itemize}
    \item $Z$, $U$, and the $d$ matrices $T_p$: $d+2$ attention calls.
    \item Remaining row operations and multiplication cost $O(nd^2)$.
    \item Intermediate entry bounds grow; accuracy requires fresh bookkeeping.
  \end{itemize}
  \papersource{Reader derivation from Definition 4.1. The printed second term on pp. 18--19 differs; numerical check in backup.}
\end{frame}

% 16
\begin{frame}{A sentinel key exposes an unnormalized row sum}
  Let $S_i=\sum_j e^{Q_i\cdot K_j}$. Add a key with score $\ln c_i$.
  \begin{center}
    Original keys have value 1; the sentinel has value 0.
  \end{center}
  \[
    o_i=\frac{S_i}{c_i+S_i},\qquad
    o_i>\tfrac12\ \Longleftrightarrow\ S_i>c_i.
  \]
  \begin{itemize}
    \item A small additive error distinguishes multiplicatively separated thresholds.
    \item Binary search on a geometric grid recovers a relative row-sum estimate.
    \item The added coordinate $\ln c_i$ may exceed the original entry bound.
  \end{itemize}
  \papersource{Lemma 5.1 and Claim 5.2, PDF pp. 25--26. Entry-bound qualification: reader verification.}
\end{frame}

% 17
\begin{frame}{An exponential sum can reveal an integer maximum}
  For integer inputs, let $M_i=\max_j a_i\cdot b_j$ and choose constant $C_0$:
  \[
    S_i=\sum_j n^{C_0(a_i\cdot b_j)},\qquad
    n^{C_0M_i}\leq S_i\leq n^{C_0M_i+1}.
  \]
  Taking a logarithm and rounding recovers $M_i$, even from a
  constant-relative-error estimate of $S_i$.
  \begin{block}{Conditional hardness with very slowly growing dimension}
    For every $\delta>0$, there is $C=C(\delta)$ such that
    $d=2^{C\log^* n}$ requires $n^{2-\delta}$ time under SETH.
  \end{block}
  Here $B=\poly(n)$ and $\varepsilon=\Theta(1)$; the enlarged sentinel bound remains polynomial.
  \papersource{Theorem 2.3, p. 9; Lemma 5.3 and Theorem 5.4, PDF pp. 26--27.}
\end{frame}

% 18
\begin{frame}{Large dimensions inherit matrix-multiplication hardness}
  Boolean vectors with scores $\ln(k)(a_i\cdot b_j)$ give
  \[
    S_i=\sum_j k^{a_i\cdot b_j},\qquad k=\frac{1}{10n}.
  \]
  \begin{columns}[t,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{No orthogonal pair}
        Every row has $S_i\leq nk=0.1$.
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{An orthogonal pair exists}
        Some row has $S_i\geq1$.
      \end{block}
    \end{column}
  \end{columns}
  A fixed sentinel with $c=1/2$ distinguishes the cases at constant accuracy.
  \begin{itemize}
    \item $B=O(\sqrt{\log n})$; input dimensions become $n+1,d+1$.
    \item Theorem 5.5: $\tmul(n,d,n)^{1-o(1)}$ under generalized high-dimensional OV.
  \end{itemize}
  \papersource{Theorem 5.5 and Eq. (5), PDF pp. 27--28. Fixed-threshold sentinel: reader simplification.}
\end{frame}

% 19
\begin{frame}{The constant-entry claim depends on delicate parameters}
  The intended encoding is an integer row sum:
  \[
    S_i=\sum_j2^{a_i\cdot b_j}.
  \]
  \begin{itemize}
    \item Nonorthogonal pairs contribute even integers; orthogonal pairs contribute 1.
    \item Independent half-sampling detects an odd count with probability $1/2$.
    \item Inverse-polynomial accuracy allows exact rounding when $d=O(\log n)$.
  \end{itemize}
  \begin{block}{Version-1 proof qualification}
    Both $Q$ and $K$ are printed with scale $\log2$, producing
    $(\log2)^2$ scores. The claimed $B=\log2$ bound also conflicts with the sentinel construction.
  \end{block}
  \papersource{Theorem 5.7 and Lemma 5.8, PDF pp. 28--29. Exact constant-bound statement not verified in this reading.}
\end{frame}

% 20
\begin{frame}{The evidence is asymptotic and conditional}
  \begin{itemize}
    \item \textbf{No empirical evaluation:} no model training, hardware benchmark, or measured speedup.
    \item \textbf{Fixed dimension:} $g^{O(d)}$ can dominate at realistic head sizes.
    \item \textbf{Worst-case guarantees:} entrywise accuracy and input assumptions define the lower bounds.
    \item \textbf{Unresolved interval:} $1\ll d\ll2^{\Theta(\log^* n)}$.
    \item \textbf{Large-$d$ optimality:} relative to fast matrix multiplication; for $d=n$, the scale is $n^\omega$, not $n^3$.
  \end{itemize}
  \papersource{Table 1, PDF p. 5; Secs. 3 and 5; Conclusion, p. 30. Practical interpretation: reader analysis.}
\end{frame}

% 21
\begin{frame}{The lasting idea is query-specific local approximation}
  \begin{enumerate}
    \item \textbf{Algorithm:} tail bounds create short score windows; geometry aggregates polynomial moments.
    \item \textbf{Complexity:} fixed dimension permits truly subquadratic attention with polylogarithmic magnitude dependence.
    \item \textbf{Hardness:} exponential weighting connects attention to integer Max-IP and OV.
    \item \textbf{Reading discipline:} preserve the hypotheses and verify extensions before using their strongest claims.
  \end{enumerate}
  \begin{block}{Discussion}
    Can the local-window idea produce a practical algorithm with a useful,
    explicitly stated accuracy guarantee?
  \end{block}
  \papersource{Main algorithm: Theorem 1.3; reductions: Sec. 5. Discussion question: reader analysis.}
\end{frame}

% 22
\begin{frame}{Sources and reading coverage}
  \textbf{Primary paper}
  \par\smallskip
  Gupta, Huang, Saha, Xu, Ye.
  \emph{Subquadratic Algorithms and Hardness for Attention with Any Temperature}.
  \href{https://arxiv.org/abs/2505.14840v1}{arXiv:2505.14840v1}.
  \par\medskip
  \begin{itemize}
    \item Read all substantive sections, PDF pp. 1--30, using author TeX and PDF.
    \item Figures 1--2 are original PDF crops; Table 1 was checked.
    \item Core tools cited by the paper: Matou\v{s}ek (1992), Aggarwal--Alman (2022), Chen (2018).
    \item External normalization check only:
      Alman--Song, \href{https://arxiv.org/abs/2302.13214v2}{arXiv:2302.13214v2}.
  \end{itemize}
  \papersource{Reader analysis and numerical checks are explicitly labeled. No independent implementation or benchmark is claimed.}
\end{frame}

\appendix

% 23
\begin{frame}{Backup: normalization contributes the second error term}
  For retained probabilities,
  \[
    \frac{1-\eta}{1+\eta}p^{(r)}_{ij}
    \leq\widehat p_{ij}\leq
    \frac{1+\eta}{1-\eta}p^{(r)}_{ij}.
  \]
  For $\eta<0.1$, this implies
  \[
    |\widehat O_{i,t}-O^{(r)}_{i,t}|
    \leq B\sum_{j\in R_i}|\widehat p_{ij}-p^{(r)}_{ij}|
    \leq4\eta B.
  \]
  \begin{itemize}
    \item Add the tail bound $3\eta B$ to obtain $7\eta B$.
    \item Signed values require an additive-output argument.
    \item With $\eta\asymp\varepsilon/B$, $g=O(\log(nB/\varepsilon))$.
  \end{itemize}
  \papersource{Sec. 3.1, Eq. (1), PDF p. 11; Lemma 3.10, p. 14.}
\end{frame}

% 24
\begin{frame}{Backup: locate a maximum without enumerating scores}
  Find the largest integer $s_i$ such that the halfspace
  \[
    \{x:Q_i\cdot x\geq s_i\log(1+\eta)\}
  \]
  contains a key. Since $|Q_i\cdot K_j|\leq dB^2$, binary search needs
  $O(\log(dB/\eta))$ count queries.
  \[
    c_i=s_i\log(1+\eta)-\log(n/\eta).
  \]
  \begin{itemize}
    \item Relevant shifted scores lie in
      $[0,\log(n/\eta)+\log(1+\eta)]$.
    \item Reuse the same halfspace for every weighted moment.
    \item A zero query can return the value mean directly.
  \end{itemize}
  \papersource{Definition 3.8 and Lemma 3.12, PDF pp. 13, 15--16. Zero-query handling: reader implementation note.}
\end{frame}

% 25
\begin{frame}{Backup: what is missing from the low-rank proof?}
  \begin{itemize}
    \item Printed product: $V_Q^\top U_KV_K^\top$.
      The first two factors have dimensions $r_Q\times d$ and $n\times r_K$.
    \item The proof treats $\max(r_Q,r_K)$ as constant, while the theorem assumes only their minimum is constant.
    \item With $V\in\R^{n\times d}$, the moment construction still has $d$ output channels.
  \end{itemize}
  \begin{block}{A conservative direct extension}
    One-sided score reduction plus separate output channels gives a bound of the form
    $\widetilde O(nd+d\,n^{2-1/r}\polylog(B/\varepsilon))$.
  \end{block}
  A stronger batched result would need an additional argument.
  \papersource{PDF pp. 16--17. Dimension check and conservative bound: reader analysis, not a lower bound or an author correction.}
\end{frame}

% 26
\begin{frame}{Backup: the printed gradient fails a small algebra check}
  $n=2,d=1$, $A_1=(1,2)^\top$, $A_2=(-1,1)^\top$,\par
  $H=(1,3)^\top$, $E=(0.2,-0.1)^\top$, and $X=0.4$.
  \begin{center}
    \begin{tabular}{@{}lr@{}}
      \toprule
      Calculation & Derivative \\
      \midrule
      Centered finite difference ($h=10^{-5}$) & $4.9557469640$ \\
      Correct row-softmax identity & $4.9557469642$ \\
      Printed identity, PDF pp. 18--19 & $-0.4520133397$ \\
      \bottomrule
    \end{tabular}
  \end{center}
  \[
    \text{Correct subtraction: }\diag((F\circ CH^\top)\one)F.
  \]
  \begin{itemize}
    \item Finite differences also agree at $h=10^{-4}$ and $10^{-6}$.
    \item This checks an identity, not the impossibility of a gradient algorithm.
  \end{itemize}
  \papersource{Reader's reproducible example: build/verify\_math.py and build/math-check.json. Not a paper experiment.}
\end{frame}

% 27
\begin{frame}{Backup: parameter changes matter in hardness reductions}
  \begin{itemize}
    \item \textbf{Generic sentinel:} $\ln c_i$ can have magnitude
      $O(\log n+dB^2)$; the original bound $B$ is not automatically preserved.
    \item \textbf{Polynomial-entry Max-IP:} this enlargement stays polynomial,
      preserving the relevant parameter regime.
    \item \textbf{High-dimensional OV:} a fixed threshold suffices;
      $S\leq0.1$ gives $o\leq1/6$, while $S\geq1$ gives $o\geq2/3$.
    \item \textbf{Constant-entry parity:} using $\sqrt{\log2}$ on both sides
      fixes integer scores but exceeds the printed $B=\log2$ threshold.
  \end{itemize}
  \papersource{Lemma 5.1, Theorems 5.5 and 5.7, PDF pp. 25--29. Parameter audit and fixed-threshold repair: reader analysis.}
\end{frame}

\end{document}