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\begin{document}

\title{Supporting Information for ``Efficient Training of the Memristive Deep
Belief Net Immune to Non-Idealities of the Synaptic Devices''}



\author[1]{Wei Wang}%
\author[1]{Barak Hoffer}%
\author[1]{Tzofnat Greenberg-Toledo}%
\author[1]{Yang Li}%
\author[1]{Minhui Zou}%
\author[1]{Eric Herbelin}%
\author[1]{Ronny Ronen}%
\author[1]{Xiaoxin Xu}%
\author[1]{Yulin Zhao}%
\author[1]{Jianguo Yang}%
\author[1]{Shahar Kvatinsky}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 29, 2024}


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\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure S1.} a) The memristive synaptic behavior with an ideally
symmetric and linear weight update ability (constant \selectlanguage{greek}Δ\selectlanguage{english}G for identical
pulses) but limited conductance levels (\emph{N} =20). b) Test accuracy
for 10,000 images in the MNIST dataset obtained during the training of
memristive DBN as a function of the training epoch
(CD\textsubscript{th}=64).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure S2.} a) Deep belief network structure for the fine-tuning
procedure by the wake-sleep algorithm. b) Procedure of utilizing
wake-sleep algorithm for the fine tune of the deep belief net.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure S3.} Example traces of the conductance updates as the
responses of potentiation or depression voltage pulses. a) Weight update
traces of an ideal device with linear weight update ability. b, c)
Weight update responds of an ideal device with non-linear weight update
behavior. d) Potentiation (positive) or depression (negative) pulse
applied on the devices.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure S4.} a) Test accuracy (50 times sampling) as a function
of the training epoch for different CDth using the symmetric and linear
weight update behavior. b) Test accuracy as a function of the training
epoch for different numbers of conductance levels,
N=N\textsubscript{p}=N\textsubscript{d}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure S5.} a) Modelled weight update behavior with different
degrees of non-linearities (assuming the degrees of non-linearity in the
potentiation and depression are the same). b) Test accuracy as a
function of the training epoch for different degrees of symmetric
nonlinearities of the weigh updates. c) Modelled weight update behavior
with different degrees of non-linearities (assuming the degree of
non-linearity in depression is constant, i.e., asymmetric weigh
updates). d) Test accuracy as a function of the training epoch for
different degrees of asymmetric nonlinearities of the weigh updates.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure S6.} a) Modelled weight update behavior with different
degrees of cycle-to-cycle variations. b) Test accuracy as a function of
the training epoch for different degrees of cycle-to-cycle variations.
c) Modelled weight update behavior with different degrees of
device-to-device variations. d) Test accuracy as a function of the
training epoch for different degrees of device-to-device variations.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure S7.} a) Test accuracy as a function of the training epoch
for various yields of the memristive device in the array. b) Test
accuracy as a function of the training epoch for various read noise
levels of the memristive device.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure S8.} a) For the device that only has gradual conductance
incremental behavior (PCM device in Fig. 5e), potentiation and
depression operation are performed in the positive branch and negative
branch of the synaptic cell consisting of a differential pair of the
memristive devices, respectively. b) Illustration of typical synaptic
evolution of the differential pair. The two memristive devices are
initialized to low conductance status. Increment of the conductance of
the positive branch results in the gradual increase of the overall
synaptic weight. Vice-versa increment the conductance of the negative
branch results in the gradual decrease of the overall synaptic weight.
However, after multiple weight update operations, the differential pair
might reach a status that both the devices are in high conductance
states, which prevents further weight update operations and degrades the
performance of the neural network. In the training case in this work, no
such issues were detected, since the number of weight update operations
on each memristive synaptic cell is highly reduced. c) For the device
only has gradual conductance decrement behavior (OxRRAM device in Fig.
5d), potentiation and depression operation are performed in the negative
branch and positive branch of the synaptic cell consisting of a
differential pair of the memristive devices, respectively. b)
Illustration of typical synaptic evolution of the differential pair.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

\textbf{Figure S9.} Test accuracy as a function of the training epoch
when using various device data to train the neural network.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

\textbf{Figure S10.} a) Schematic of using the trained DBN to generate
handwritten digit images by only providing the labels. b) Generated
handwritten digit images by providing only the labels.

\textbf{Table~S1.} The fitting parameter values for the various
memristive synaptic behaviors.\selectlanguage{english}
\begin{longtable}[]{@{}lllllllll@{}}
\toprule
Device & \emph{G\textsubscript{max}} & \emph{G\textsubscript{min}} &
\emph{N\textsubscript{P}} & \emph{N\textsubscript{d}} &
\selectlanguage{greek}\emph{α\textsubscript{p}}\selectlanguage{english} & \selectlanguage{greek}\emph{α\textsubscript{d}}\selectlanguage{english} & \selectlanguage{greek}\emph{γ} &
Accuracy\tabularnewline
\midrule
\endhead
SiGe epiRAM {[}9{]} & 40 \selectlanguage{greek}µ\selectlanguage{english}S & 1 \selectlanguage{greek}µ\selectlanguage{english}S & 500 & 400 & 8 & 15 & 2 &
95.01\%\tabularnewline
& 30 \selectlanguage{greek}µ\selectlanguage{english}S & 0.1 \selectlanguage{greek}µ\selectlanguage{english}S & 200 & 50 & 5 & 1 & 1 & 95.80\%\tabularnewline
& 12.5 \selectlanguage{greek}µ\selectlanguage{english}S & 0.1 \selectlanguage{greek}µ\selectlanguage{english}S & 100 & 50 & 1 & 1 & 1 & 95.76\%\tabularnewline
PCMO {[}10{]} & 0.16 \selectlanguage{greek}µ\selectlanguage{english}S & 35 nS & 100 & 100 & 6 & 20 & 1 &
95.78\%\tabularnewline
ECRAM {[}6{]} & 3 nS & 1 nS & 55 & 55 & 0.5 & 0.5 & 0.3 &
95.24\%\tabularnewline
OxRRAM {[}11{]} & 250 \selectlanguage{greek}µ\selectlanguage{english}S & 20 \selectlanguage{greek}µ\selectlanguage{english}S & - & 100 & - & 15 & 0.3 &
96.37\%\tabularnewline
PCM {[}12{]} & 2.2 mS & 7 \selectlanguage{greek}µ\selectlanguage{english}S & 30 & - & 6 & - & 0.3 &
96.96\%\tabularnewline
\bottomrule
\end{longtable}

\selectlanguage{english}
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\end{document}

