# HYPER-SNN: Towards Energy-efficient Quantized Deep Spiking Neural Networks
for Hyperspectral Image Classification

Gourav Datta, Souvik Kundu, Akhilesh R. Jaiswal, Peter A. Beerel G. Datta, S.
Kundu, A. R. Jaiswal and P. A. Beerel are with the Department of Electrical
and Computer Engineering, University of Southern California, Los Angeles, CA,
90089 USA e-mail: {gdatta, souvikku, akhilesh<EMAIL_ADDRESS>
###### Abstract

Hyperspectral images (HSIs) provide rich spectral–spatial information across a
series of contiguous spectral bands. However, the accurate processing of the
spectral and spatial correlation between the bands requires the use of energy-
expensive 3-D Convolutional Neural Networks (CNNs). To address this challenge,
we propose the use of Spiking Neural Networks (SNNs) that are generated from
iso-architecture CNNs and trained with quantization-aware gradient descent to
optimize their weights, membrane leak, and firing thresholds. During both
training and inference, the analog pixel values of a HSI are directly applied
to the input layer of the SNN without the need to convert to a spike-train.
The reduced latency of our training technique combined with high activation
sparsity yields significant improvements in computational efficiency. We
evaluate our proposal using three HSI datasets on a 3-D and a 3-D/2-D hybrid
convolutional architecture. We achieve overall accuracy, average accuracy, and
kappa coefficient of $98.68\%$, $98.34\%$, and $98.20\%$ respectively with $5$
time steps (inference latency) and $6$-bit weight quantization on the Indian
Pines dataset. In particular, our models achieved accuracies similar to state-
of-the-art (SOTA) with ${\sim}560.6\times$ and ${\sim}44.8\times$ less compute
energy on average over three HSI datasets than an iso-architecture full-
precision and 6-bit quantized CNN, respectively.

###### Index Terms:

hyperspectral images, spiking neural networks, quantization-aware, gradient
descent, indian pines

##  I Introduction

Hyperspectral imaging, which extracts rich spatial-spectral information about
the ground surface, has shown immense promise in remote sensing [1]. It is
currently used in several applications ranging from geological surveys [2], to
the detection of camouflaged vehicles [3]. In hyperspectral images (HSIs),
each pixel can be considered as a high-dimensional vector where each entry
corresponds to the spectral reflectivity [1] of a particular wavelength. The
goal of the classification task is to assign a unique semantic label to each
pixel [4].

For HSI classification, several spectral feature-based methods have been
proposed, including support vector machine [5], random forest [6], canonical
correlation forest [7], and multinomial logistic regression [8]. To improve
the accuracy of HSI classification, researchers have integrated spatial
features into existing learning methods [9]. Some spectral-spatial methods for
classifying HSIs include fusing correlation coefficient and sparse
representation [10], Boltzmann entropy-based band selection [11], joint sparse
model and discontinuity preserving relaxation [12], and extended morphological
profiles [13, 14]. Some of these methods have also been proposed to exploit
the spatial context with various morphological operations for HSI
classification. However, these spectral-spatial feature extraction methods
rely on hand-designed descriptions, prior information, and empirical
hyperparameters [1].

Lately, convolutional neural networks (CNNs) have yielded higher accuracy than
some hand-designed features [15]. CNNs have shown promise in multiple
applications where visual information processing is required, including image
classification [16], object detection [17], semantic segmentation [18], and
depth estimation [19]. In particular, CNN-based methods act as an end-to-end
feature extractor that consists of a series of hierarchical filtering layers
for global optimization. The 2-D CNN stacked autoencoder [1] was the first
attempt to extract deep features from its compressed latent space to classify
HSIs. Following this work, [20] employed a 2-D CNN model to extract the
spatial information in a supervised manner and classify the raw hyperspectral
images. The multibranch selective kernel network with attention [21] and
pixel-block pair based data augmentation techniques [22] were developed to
address the gradient vanishing and overfitting problems. To extract the
spatial-spectral features jointly from the raw HSI, researchers proposed a 3-D
CNN architecture [23], which achieves even better classification results.
Authors in [24, 25, 26] created multiscale spatiospectral relationships using
3-D CNN and fused the features using a 2-D CNN to extract more robust
representation of spectral–spatial information.

However, the performance and success of multi-layer CNNs are generally
associated with high power and energy costs [27]. A typical hyperspectral
image cube consists of several hundred spectral frequency bands, and hence,
classifying these images using traditional CNNs require a large amount of
computational power, especially when real time processing is necessary, as in
target tracking or identification [28]. The high energy cost and the demand
for deployment of HSI sensors in battery-powered edge devices motivates
exploring alternative lightweight energy-efficient HSI classification models.

In particular, low-latency spiking neural networks (SNNs) [29] have gained
attention because they can be more computational efficient than CNNs for a
variety of applications, including image analysis. To achieve this goal,
analog inputs are first encoded into a sequence of spikes using one of a
variety of proposed encoding methods, including rate coding [30, 31], direct
coding [32], temporal coding [33], rank-order coding [34], phase coding [35],
and other exotic coding schemes [36, 37]. Among these, rate and direct coding
have shown competitive performance on complex tasks [30, 31] while others are
either limited to simpler tasks such as learning the XOR function and
classifying MNIST images or require a large number of spikes for inference.

In particular, for rate coding, the analog value is converted to a spike train
using a Poisson generator function with a rate proportional to the input pixel
value. The number of timesteps $T$ in each train is inversely proportional to
the quantization error in the representation, as illustrated in Fig. 1(b).
[31]. In contrast, in direct-input encoding, the analog pixel values are fed
into the first convolutional layer as multi-bit values that are fixed for all
$T$ timesteps. [32].

In addition to accommodating various forms of encoding inputs, supervised
learning algorithms for SNNs have overcome various roadblocks associated with
the discontinuous derivative of the spike activation function [38, 39]. In
particular, recent works have shown that SNNs can be efficiently converted
from artifical neural networks (ANNs) by approximating the activation value of
ReLU neurons with the firing rate of spiking neurons [31]. Low-latency SNNs
trained using ANN-SNN conversion, coupled with supervised training, have been
able to perform at par with ANNs in terms of classification accuracy in
traditional image classification tasks [32]. This motivates this work which
explores the effectiveness of SNNs for HSI classification.

More specifically, this paper provides the following contributions:

  * •

We propose two convolutional architectures for HSI classification that can
yield classification accuracies similar to state-of-the-art (SOTA) and are
compatible with our ANN-SNN conversion framework.

  * •

We propose a hybrid training algorithm that first converts an ANN for HSI
classification to an iso-architecture SNN, and then trains the latter using a
novel quantization-aware spike timing dependent backpropagation (Q-STDB)
algorithm.

  * •

We evaluate and compare the energy-efficiency of the SNNs obtained by our
training framework, with standard ANNs, using appropriate energy models, which
reveal that our SNNs trained for HSI classification can offer significant
improvement in compute efficiency.

The remainder of this paper is structured as follows. In Section II we present
necessary background and related work. Section III and IV describe our
quantization-aware SNN training method and network architectures respectively.
We present the detailed experimental evaluations of our proposal in Section V.
We show the improvement in energy-efficiency of our proposed SNN for all the
HSI classification tasks in Section VI. Finally, the paper concludes in
Section VII.

##  II Background and Related Work

###  II-A SNN Modeling

An SNN consists of a network of neurons that communicate via a sequence of
spikes modulated by synaptic weights. The activity of pre-synaptic neurons
modulates the membrane potential of postsynaptic neurons, generating an action
potential or spike when the membrane potential crosses a firing threshold. The
spiking dynamics of a neuron are generally modeled using either the Integrate-
and-Fire (IF) [40] or Leaky-Integrate-and-Fire (LIF) model [41]. Both IF and
LIF neurons accumulate the input current into their respective states or
membrane potentials. The difference between the two models is that the
membrane potential of a IF neuron does not change during the time period
between successive input spikes while the LIF neuronal membrane potential
leaks at a constant rate. In this work, we use the LIF model to convert ANNs
trained with ReLU activations, to SNNs, because the leak term provides a
tunable control knob that can reduce inference latency and spiking
activity.The IF model can be characterized by the following differential
equation

Figure 1: (a) Feedforward fully-connected SNN architecture with integrate and
fire (IF) spiking dynamics, (b) The spike input generated over several
timesteps through a Poisson generator. It is clear that the larger the number
of timesteps, the better the accumulated input spikes approximates the
original input image.

$C\frac{dU_{i}(t)}{dt}=I_{i}(t)=\sum_{j}W_{ij}\cdot{S_{j}(t)}$ (1)

where $C$ is the membrane capacitance, $U_{i}(t)$ and $I_{i}(t)$ are the
membrane potential and input synaptic current of the $i^{th}$ neuron at time
$t$. As illustrated in Fig. 1(a), $U_{i}(t)$ integrates the incoming (pre-
neuron) binary spikes $S_{j}(t)$ multiplied by weights $W_{ij}$. The neuron
generates an output spike when $U_{i}$ exceeds the firing threshold $V$.
However, because of its continuous-time representation, Eq. 1 is incompatible
for implementation in common Machine Learning (ML) frameworks (e.g. Pytorch).
Hence, we follow an iterative version evaluated in discrete time, within which
spikes are characterized as binary values (1 represents the presence of a
spike) [42].

$U_{i}(t)=U_{i}(t-1)+\sum_{j}W_{ij}{S_{j}(t)}-{V}\cdot{O_{j}(t-1)}$ (2)

$O_{i}(t-1)=\begin{cases}1,&\text{if }U_{i}(t-1)>V\\\
0,&\text{otherwise}\end{cases}$ (3)

$O_{i}(t)$ is the output spike at time step $t$. Note that the third term in
Eq. 2 exhibits soft reset by reducing the membrane potential $U_{i}$ by the
threshold $V$ at time step $t$, if an output spike is generated at the
$(t-1)^{th}$ time step. Alternatively, hard reset implies resetting $U_{i}$ to
$0$. Soft reset minimizes the information loss by allowing the spiking neuron
to carry forward the surplus potential above the firing threshold to the next
time step [42].

###  II-B SNN Training Techniques

Recent research on training supervised deep SNNs can be broadly divided into
three categories: 1) ANN-to-SNN conversion-based training, 2) Spike timing
dependent backpropagation (STDB), and 3) Hybrid training.

####  II-B1 ANN-to-SNN Conversion

Recent works have demonstrated that SNNs can be efficiently converted from
ANNs by approximating the activation value of ReLU neurons with the firing
rate of spiking neurons [43, 44, 45, 31, 46]. This technique uses standard
backpropagation-based training for the ANN models and helps an iso-
architecture SNN achieve superior classification accuracy in image recognition
tasks [44, 31]. However, the SNNs resulting from these ANN-SNN conversion
algorithms require an order of magnitude higher latency compared to other
training techniques [31]. In this work, we use ANN-SNN conversion as an
initial step in Q-STDB because it is of relatively low complexity and yields
high classification accuracy on deep networks.

####  II-B2 STDB

The threshold comparator in the IF neuronal model yields a discontinuous and
thus non-differentiable function, making it incompatible with the powerful
gradient-descent based learning methods. Consequently, several approximate
training methodologies have been proposed to overcome the challenges
associated with non-differentiability [38, 47, 48, 49]. The key idea of these
works is to approximate the spiking neuron functionality with a continuous
differentiable model or use surrogate gradients as an approximate version of
the real gradients to perform gradient descent based training. Unfortunately,
SNNs trained using this approach generally require a large number of time
steps, in the order of few hundreds, to process an input. As a result, the
backpropagation step requires the gradients of the unrolled SNN to be
integrated over all these time steps. This multiple-iteration backpropagation-
through-time (BPTT) coupled with the exploding memory complexity has hindered
the applicability of surrogate gradient based learning methods to deep
convolutional architectures.

####  II-B3 Hybrid Training

A recent paper [42] proposed a hybrid training methodology where the ANN-SNN
conversion is performed as an initialization step and is followed by an
approximate gradient descent algorithm. The authors observed that combining
the two training techniques helps the SNNs converge within a few epochs while
requiring fewer time steps. Another recent paper [32] proposed a training
scheme for deep SNNs in which the membrane leak and the firing threshold along
with other network parameters (weights) are updated at the end of every batch
via gradient descent after ANN-SNN conversion. Moreover, [32] applied direct-
input encoding where the pixel intensities of an image are fed into the SNN
input layer as fixed multi-bit values each timestep to reduce the number of
required fewer time steps needed to achieve SOTA accuracy. Thus, the first
convolutional layer composed of LIF neurons acts as both a feature extractor
and spike-generator. This is similar to rate-coding except that the spike-rate
of the first hidden layer is a function of its weights, membrane leak, and
threshold parameters that are all learned by gradient descent. This work
extends these hybrid learning techniques by incorporating weight quantization,
as defined below.

##  III Proposed Quantized SNN Training Method

In this section, we evaluate and compare the different choices for SNN
quantization in terms of compute efficiency and model accuracy. We then
incorporate the chosen quantization technique into STDB, which we refer to as
Q-STDB.

###  III-A Study of Quantization Choice

Uniform quantization transforms a weight element $w\in[w_{min},w_{max}]$ to a
range $[-2^{b-1},2^{b-1}-1]$ where $b$ is the bit-width of the quantized
integer representation. There are primarily two choices for the above
transformation, known as affine and scale quantization. Detailed descriptions
of these two types of quantization can be found in [50].

Our key motivation for SNN weight quantization is the hardware acceleration of
inference using energy-efficient integer or fixed-point computational units
implemented as crossbar array based processing-in-memory (PIM) accelerators.
Note that the six transistor SRAM array based in-memory computing requires
low-precision weights for multiply-and-accumulate (MAC) operations due to low
density of the bit-cells. Previous research [51, 52] have proposed post-
training SNN quantization tailored towards unsupervised learning, which may
not scale to complex vision tasks without requiring high-precision ($\geq{8}$
bits). In contrast, in this work, we propose quantization-aware training,
where the weights are fake quantized (see [50]) in the forward path
computations, while the gradients and weight updates are calculated using the
full precision weights. There are several choices for sharing quantization
parameters among the tensor elements in a SNN. We refer to this choice as
quantization granularity. We employ per-tensor (or per-layer) granularity
where the same quantization parameters are shared by all elements in the
tensor, because this reduces the computational cost compared to other
granularity choices with no impact on model accuracy. Activations are
similarly quantized, but only in the SNN input layer, since they are binary
spikes in the remaining layers.

To evaluate the compute cost, let us consider a 3-D convolutional layer $l$,
the dominant layer in HSI classification models, that performs a tensor
operation $O_{l}=X_{l}\circledast W_{l}$ where
$X_{l}\in\mathbb{R}^{H_{l}^{i}\times{W_{l}^{i}}\times{C_{l}^{i}}\times{D_{l}^{i}}}$
is the input activation tensor,
$W_{l}\in\mathbb{R}^{k_{l}^{x}\times{k_{l}^{y}}\times{k_{l}^{z}}\times{C_{l}^{i}}\times{C_{l}^{o}}}$
and
$O^{l}\in\mathbb{R}^{H_{l}^{o}\times{W_{l}^{o}}\times{C_{l}^{o}}\times{D_{l}^{o}}}$
is the output activation tensor, where $H_{l}^{i}$, $W_{l}^{i}$, $C_{l}^{i}$,
$D_{l}^{i}$ are the input height, width, channel size, and spectral size,
respectively. Similarly, $H_{l}^{o}$, $W_{l}^{o}$, $C_{l}^{o}$ and $D_{l}^{o}$
are the output height, width, number of filters and the output spectral size,
respectively, and $k_{l}^{x}$, $k_{l}^{y}$, $k_{l}^{z}$ represent the filter
size in the three spatial dimensions. The result of the real-valued operation
$O_{l}=X_{l}\circledast W_{l}$ can be approximated with quantized tensors
$X_{l}^{Q}$ and $W_{l}^{Q}$, by first dequantizing them producing
$\hat{X_{l}}$ and $\hat{W_{l}}$ respectively, and then performing the
convolution. Note that both $X_{l}^{Q}$ and $W_{l}^{Q}$ have similar
dimensions as $X_{l}$ and $W_{l}$ respectively. Assuming the tensors are
scale-quantized per layer,

$O_{l}=X_{l}\circledast
W_{l}\approx\hat{X_{l}}\circledast\hat{W_{l}}={X_{l}^{Q}\circledast
W_{l}^{Q}}\cdot(\frac{1}{s_{s}^{X}\cdot{s_{s}^{W}}})$ (4)

where $s_{s}^{X}$ and $s_{s}^{W}$ are scalar values for scale quantization
representing the levels of the input and weight tensor respectively. Hence,
scale quantization results in an integer convolution, followed by a point-wise
floating-point multiplication for each output element. Given that a typical
convolution operation involves a few hundred MAC operations (accumulate for
binary spike inputs) to compute an output element, a single floating-point
operation for the scaling shown in Eq. 4 is a negligible computational cost.
Note that $X_{l}$ only needs to be quantized if $l$ is the input layer. In all
other cases, $X_{l}^{Q}=X_{l}$ and $s_{s}^{X}=1$.

Although both affine and scale quantization enable the use of low-precision
arithmetic, affine quantization results in higher computationally expensive
inference as shown below.

$\displaystyle O_{l}$
$\displaystyle\approx\frac{X_{l}^{Q}-z_{a}^{X}}{s_{a}^{X}}\circledast\frac{W_{l}^{Q}-z_{a}^{W}}{s_{a}^{W}}$
$\displaystyle=\frac{({X_{l}^{Q}}\circledast{W_{l}^{Q}}-z_{a}^{X}\circledast(W_{l}^{Q}-z_{a}^{W})-X_{l}^{Q}\circledast
z_{a}^{W})}{s_{a}^{X}\cdot{s_{a}^{W}}}$ (5)

where $z_{a}^{X}$ and $z_{a}^{W}$ are tensors of sizes equal to that of
$X_{l}^{Q}$ and $W_{l}^{Q}$ respectively that consist of repeated elements of
the scalar zero-values of the input activation and weight tensor respectively.
On the other hand, $s_{a}^{X}$ and $s_{a}^{W}$ are the corresponding scale
values. The first term in the numerator of Eq. III-A is the integer
convolution operation similar to the one performed in scale quantization shown
in Eq. 4. The second term contains integer weights and zero-points, which can
be computed offline, and adds an element-wise addition during inference. The
third term, however, involves the quantized activation $X_{l}^{Q}$, which
cannot be computed offline. This extra computation, depending on the
implementation, can introduce considerable overhead, reducing or even
eliminating the throughput and energy advantage that low precision PIM
accelerators offer over floating-point MAC units. Hence, we use scale
quantization during inference.

Note, however, that our experiments detailed in Section V show that using
scale quantization during SNN training degrades the test accuracy
significantly. Hence, we propose that training should use affine quantization
of both the weights and input layer activations. Note that for a integer math
unit or PIM accelerator, we do not necessarily need to quantize the SNN
membrane potentials which are obtained as results of the accumulate operations
of the weight elements. This is because the membrane potentials only need to
be compared with the threshold voltage once for each time step, which consumes
negligible energy, and can be performed using high precision fixed-point
comparators (in the periphery of the crossbar array for PIM accelerators).
However, quantizing the potentials can reduce the data movement cost as
discussed in Section VI-B.

Figure 2: Proposed SNN training framework details with 3-D convolutions.

###  III-B Q-STDB based Training

In this subsection, we derive the expressions to compute the gradients of the
parameters at all layers for our training framework. Our framework, which is
illustrated in Fig. 2, incorporates the quantization methodology described
above into the STDB technique used to train SNNs [32], where the spatial and
temporal credit assignment is performed by unrolling the network in time and
employing BPTT.
Output Layer: The neuron model in the output layer $L$ only accumulates the
incoming inputs without any leakage, does not generate an output spike, and is
described by

$\displaystyle\mbox{\boldmath$u$}_{L}^{t}$
$\displaystyle=\mbox{\boldmath$u$}_{L}^{t-1}+\hat{\mbox{\boldmath$w$}_{L}}\mbox{\boldmath$o$}_{L-1}^{t}$
(6)

where $N$ is the number of output labels, $\mbox{\boldmath$u$}_{L}$ is a
vector containing the membrane potential of $N$ output neurons,
$\mbox{\boldmath$\hat{w}$}_{L}$ is the fake quantized weight matrix connecting
the last two layers ($L$ and $L{-}1$), and $\mbox{\boldmath$o$}_{L-1}$ is a
vector containing the spike signals from layer $(L{-}1)$. The loss function is
defined on $\mbox{\boldmath$u$}_{L}$ at the last time step $T$. We employ the
cross-entropy loss and compute the softmax of $\mbox{\boldmath$u$}_{L}^{T}$.
The output of the network is passed through a softmax layer that outputs a
probability distribution. The loss function $\mathcal{L}$ is defined as the
cross-entropy between the true output ($y$) and the SNN’s predicted
distribution ($p$).

Figure 3: Fake quantization forward and backward pass with straight through
estimator (STE) approximation

$\mathcal{L}=-\sum_{i=1}^{N}{y_{i}log({p_{i}})},\quad{p_{i}}=\frac{e^{u_{i}^{T}}}{\sum_{j=1}^{N}e^{u_{j}^{T}}},$
(7)

The derivative of the loss function with respect to the membrane potential of
the neurons in the final layer is described by
$\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$u$}_{L}^{T}}=(\mbox{\boldmath${p}$}-\mbox{\boldmath$y$})$,
where $p$ and $y$ are vectors containing the softmax and one-hot encoded
values of the true label respectively. To compute the gradient at the current
time step, the membrane potential at the previous step is considered as an
input quantity [32]. With the weights being fake quantized, gradient descent
updates the network parameters $\mbox{\boldmath$w$}_{L}$ of the output layer
as

$\displaystyle\mbox{\boldmath$w$}_{L}$
$\displaystyle=\mbox{\boldmath$w$}_{L}-\eta\Delta{\mbox{\boldmath{$w$}}_{L}}$
(8) $\displaystyle\Delta{\mbox{\boldmath$w$}_{L}}$
$\displaystyle=\sum_{t}\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$w$}_{L}}=\sum_{t}\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$u$}_{L}^{t}}\frac{\partial\mbox{\boldmath$u$}_{L}^{t}}{\partial\mbox{\boldmath$\hat{w}$}_{L}}\frac{\partial\mbox{\boldmath$\hat{w}$}_{L}}{\partial\mbox{\boldmath$w$}_{L}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$u$}_{L}^{T}}\sum_{t}\frac{\partial\mbox{\boldmath$u$}_{L}^{t}}{\partial\mbox{\boldmath$\hat{w}$}_{L}}\frac{\partial\mbox{\boldmath$\hat{w}$}_{L}}{\partial\mbox{\boldmath$w$}_{L}}\approx(\mbox{\boldmath${p}$}-\mbox{\boldmath$y$})\sum_{t}\mbox{\boldmath$o$}_{L-1}^{t}$
(9)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$o$}_{L-1}^{t}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$u$}_{L}^{t}}\frac{\partial\mbox{\boldmath$u$}_{L}^{t}}{\partial\mbox{\boldmath$o$}_{L-1}^{t}}=(\mbox{\boldmath${p}$}-\mbox{\boldmath$y$})\mbox{\boldmath$\hat{w}$}_{L}$
(10)

where $\eta$ is the learning rate (LR). Note that the derivative of the fake
quantization function of the weights
($\frac{\partial\mbox{\boldmath$\hat{w}$}_{L}}{\partial\mbox{\boldmath$w$}_{L}}$)
is undefined at the step boundaries and zero everywhere, as shown in Fig.
3(a). Our training framework addresses this challenge by using the Straight-
through Estimator (STE) [53], which approximates the derivative to be equal to
1 for inputs in the range $[w_{min},w_{max}]$ as shown in Fig. 3(b), where
$w_{min}$ and $w_{max}$ are the minimum and maximum values of the weights in a
particular layer. Note that $w_{min}$ and $w_{max}$ are updated at the end of
every mini-batch to ensure all the weights lie between $w_{min}$ and $w_{max}$
during the forward and backward computations in each training iteration.
Hence, we use
$\frac{\partial\mbox{\boldmath$\hat{w}$}_{L}}{\partial\mbox{\boldmath$w$}_{L}}\approx
1$ to compute the loss gradients in Eq. 9.
Hidden layers: The neurons in the hidden convolutional and fully-connected
layers are defined by the quantized LIF model as

$\displaystyle\mbox{\boldmath$u$}_{l}^{t}$
$\displaystyle=\lambda_{l}{\mbox{\boldmath$u$}_{l}^{t-1}}+\mbox{\boldmath$\hat{w}$}_{l}{\mbox{\boldmath$o$}_{l-1}^{t}}-v_{l}\mbox{\boldmath$o$}_{l}^{t-1}$
(11) $\displaystyle\mbox{\boldmath$z$}_{l}^{t}$
$\displaystyle=\frac{\mbox{\boldmath$u$}_{l}^{t}}{v_{l}}-1,\quad\mbox{\boldmath$o$}_{l}^{t}=\begin{cases}1,&\text{if
}\mbox{\boldmath$z$}_{l}^{t}>0\\\ 0,&\text{otherwise }\end{cases}$ (12)

where $\lambda_{l}$ and $v_{l}$ represent the leak and threshold potential for
all neurons in layer $l$. All neurons in a layer possess the same leak and
threshold value. This reduces the number of trainable parameters and we did
not observe any significant improvement by assigning individual threshold/leak
to each neuron. Given that the threshold is same for all neurons in a
particular layer, it may seem redundant to train both the weights and
threshold together. However, we observe that the number of time steps required
to obtain the state-of-the-art classification accuracy decreases with this
joint optimization. We hypothesize that this is because the optimizer is able
to reach an improved local minimum when both parameters are tunable. The
weight update in Q-STDB is calculated as

$\displaystyle\Delta{w_{l}}$
$\displaystyle=\sum_{t}\frac{\partial\mathcal{L}}{\partial
w_{l}}=\sum_{t}\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$z$}_{l}^{t}}\frac{\partial\mbox{\boldmath$z$}_{l}^{t}}{\partial\mbox{\boldmath$o$}_{l}^{t}}\frac{\partial\mbox{\boldmath$o$}_{l}^{t}}{\partial\mbox{\boldmath$u$}_{l}^{t}}\frac{\partial\mbox{\boldmath$u$}_{l}^{t}}{\partial\mbox{\boldmath$\hat{w}$}_{l}}\frac{\partial\mbox{\boldmath$\hat{w}$}_{l}}{\partial\mbox{\boldmath$w$}_{l}}$
$\displaystyle\approx\sum_{t}\frac{\partial\mathcal{L}}{\partial\mbox{\boldmath$z$}_{l}^{t}}\frac{\partial\mbox{\boldmath$z$}_{l}^{t}}{\partial\mbox{\boldmath$o$}_{l}^{t}}\frac{1}{v_{l}}\mbox{\boldmath$o$}_{l-1}^{t}\cdot{1}$
(13)

where
$\frac{\partial\mbox{\boldmath$\hat{w}$}_{l}}{\partial\mbox{\boldmath$w$}_{l}}$
and
$\frac{\partial\mbox{\boldmath$z$}_{l}^{t}}{\partial\mbox{\boldmath$o$}_{l}^{t}}$
are the two discontinuous gradients. We calculate the former using STE
described above, while the latter is approximated using surrogate gradient
[48] shown below.

$\displaystyle\frac{\partial\mbox{\boldmath$z$}_{l}^{t}}{\partial\mbox{\boldmath$o$}_{l}^{t}}=\gamma\cdot{max(0,1-|\mbox{\boldmath$z$}_{l}^{t}|)}$
(14)

Note that $\gamma$ is a hyperparameter denoting the maximum value of the
gradient. The threshold and leak update is computed similarly using BPTT [32].

##  IV Proposed Architectures

We developed two models, a 3-D and a hybrid fusion of 3-D and 2-D
convolutional architectures, that are inspired by the recently proposed CNN
models [23, 26, 25] used for HSI classification and compatible with our ANN-
SNN conversion framework. We refer to the two models CNN-3D and CNN-32H. The
models are trained without the bias term because it complicates parameter
space exploration which increases the conversion difficulty and tends to
increase conversion loss. The absence of the bias term implies that batch
normalization [54] cannot be used as a regularizer during the training
process. Instead, we use dropout [55] as the regularizer for both ANN and SNN
training. Moreover, our models employ ReLU nonlinearity after each
convolutional and linear layer (except the classifier layer) to further
decrease the conversion loss due to the similarity between ReLU and LIF
neurons. Also, our pooling operations use average pooling because for binary
spike based activation layers, max pooling incurs significant information
loss. Additionally, we modified the number of channels and convolutional
layers to obtain a reasonable tradeoff between accuracy and compute
efficiency. 2-D patches of sizes $5{\times}5$ and $3{\times}3$ were extracted
for CNN-3D and CNN-32H respectively, without any reduction in dimensionality
from each dataset. Higher sized patches increase the computational complexity
without any significant improvement in test accuracy. Our model architectures
are explicitly described in Table I.

##  V Experiments

###  V-A Datasets

We used three publicly available datasets, namely Indian Pines, Pavia
University, and Salinas scene. A brief description follows for each one [56].

Indian Pines: The Indian Pines (IP) dataset consists of $145{\times}145$
spatial pixels and $220$ spectral bands in a range of $400-2500$ nm. It was
captured using the AVIRIS sensor over North-Western Indiana, USA, with a
ground sample distance (GSD) of $20$ m and has $16$ vegetation classes.
Pavia University: The Pavia University (PU) dataset consists of hyperspectral
images with $610{\times}340$ pixels in the spatial dimension, and $103$
spectral bands, ranging from $430$ to $860$ nm in wavelength. It was captured
with the ROSIS sensor with GSD of $1.3$ m over the University of Pavia, Italy.
It has a total of 9 urban land-cover classes.
Salinas Scene: The Salinas Scene (SA) dataset contains images with
$512{\times}217$ spatial dimension and $224$ spectral bands in the wavelength
range of $360$ to $2500$ nm. The $20$ water absorbing spectral bands have been
discarded. It was captured with the AVIRIS sensor over Salinas Valley,
California with a GSD of $3.7$ m. In total $16$ classes are present in this
dataset.

For preprocessing, images in all the data sets are normalized to have a zero
mean and unit variance. For our experiments, all the samples are randomly
divided into two disjoint training and test sets. The limited 40% samples are
used for training and the remaining 60% for performance evaluation.

TABLE I: Model architectures employed for CNN-1 and CNN-2. Every convolutional and linear layer is followed by ReLU non-linearity. The last classifier layer is not shown. Layer | Number of | Size of | Stride | Padding | Dropout
---|---|---|---|---|---
type | filters | each filter | value | value | value
Architecture : CNN-3D
3-D Convolution | 20 | (3,3,3) | (1,1,1) | (0,0,0) | -
3-D Convolution | 40 | (3,1,1) | (2,1,1) | (1,0,0) | -
3-D Convolution | 84 | (3,3,3) | (1,1,1) | (1,0,0) | -
3-D Convolution | 84 | (3,1,1) | (2,1,1) | (1,0,0) | -
3-D Convolution | 84 | (3,1,1) | (1,1,1) | (1,0,0) | -
3-D Convolution | 84 | (2,1,1) | (2,1,1) | (1,0,0) | -
Architecture : CNN-32H
3-D Convolution | 90 | (18,3,3) | (7,1,1) | (0,0,0) | -
2-D Convolution | 64 | (3,3) | (1,1) | (0,0) | -
2-D Convolution | 128 | (3,3) | (1,1) | (0,0) | -
Avg. Pooling | - | (4,4) | (4,4) | (0,0) | -
Dropout | - | - | - | - | 0.2
Linear | 30998528 | - | - | - | -

###  V-B Experimental Setup

####  V-B1 ANN Training

We performed full-precision ANN training for $100$ epochs using the standard
SGD optimizer with an initial learning rate (LR) of $0.01$ that decayed by a
factor of 0.1 after $60$, $80$, and $90$ epochs.

####  V-B2 Conversion and SNN Training

We first examine the distribution of the neuron input values over the total
number of time steps across all neurons of the first layer for a small batch
of HSI images (of size $50$ in our case) and set the layer threshold to the
$99.7$ percentile of the scaled value of the evaluated threshold [32]. In our
experiments we scale the initial thresholds by 0.8. Similarly, we then compute
the thresholds of the subsequent layers sequentially by examining the
distribution of their input values. Note that we use $100$ time steps to
evaluate the thresholds, while the SNN training and inference are performed
with only $5$ time steps. We keep the leak of each layer set to unity while
evaluating initial thresholds. At the start of SNN training, we initialize the
weights with those from the trained ANN and initialize the leak parameters to
$1.0$. We then perform the quantization-aware SNN training described in
Section III for another $100$ epochs. We set $\gamma$ = $0.3$ [48] and used
the ADAM optimizer with a starting LR of $10^{-4}$ which decays by a factor of
$0.5$ after $60$, $80$, and $90$ epochs.

Figure 4: (a) Test accuracies for affine and scale quantization with CNN-3D over IP dataset (b) Test accuracies with 6, 9 and 12-bit weight precisions for post-training quantization with CNN-32H on IP dataset. TABLE II: Model performances with Q-STDB based training on IP, PU, and SS datasets for CNN-3D and CNN-32H after a) ANN training, b) ANN-to-SNN conversion, c) 32-bit SNN training, d) 4-bit SNN training, e) 5-bit SNN training, and f) 6-bit SNN training. | A. ANN | B. Accuracy after | C. Accuracy after | D. Accuracy after | E. Accuracy after | F. Accuracy after
---|---|---|---|---|---|---
Dataset |  accuracy ($\%$) |  ANN-to-SNN conv. ($\%$) |  FP SNN training ($\%$) |  4-bit SNN training (%) |  5-bit SNN training (%) |  6-bit SNN training (%)
| OA | AA | Kappa | OA | AA | Kappa | OA | AA | Kappa | OA | AA | Kappa | OA | AA | Kappa | OA | AA | Kappa
Architecture : CNN-3D
IP | 98.86 | 98.42 | 98.55 | 57.68 | 50.88 | 52.88 | 98.92 | 98.76 | 98.80 | 97.08 | 95.64 | 95.56 | 98.38 | 97.78 | 98.03 | 98.68 | 98.34 | 98.20
PU | 99.69 | 99.42 | 99.58 | 91.16 | 88.84 | 89.03 | 99.47 | 99.06 | 99.30 | 98.21 | 97.54 | 97.75 | 99.26 | 98.48 | 98.77 | 99.50 | 99.18 | 99.33
SS | 98.89 | 98.47 | 98.70 | 81.44 | 76.72 | 80.07 | 98.49 | 97.84 | 98.06 | 96.47 | 93.16 | 94.58 | 97.25 | 95.03 | 95.58 | 97.95 | 97.09 | 97.43
Architecture : CNN-32H
IP | 97.60 | 97.08 | 97.44 | 70.88 | 66.56 | 67.89 | 97.27 | 96.29 | 96.35 | 96.63 | 95.81 | 95.89 | 97.23 | 96.08 | 96.56 | 97.45 | 96.73 | 96.89
PU | 99.50 | 99.09 | 99.30 | 94.96 | 90.12 | 93.82 | 99.38 | 98.83 | 99.13 | 99.17 | 98.41 | 98.68 | 99.25 | 98.84 | 98.86 | 99.35 | 98.88 | 98.95
SS | 98.88 | 98.39 | 98.67 | 88.16 | 84.19 | 85.28 | 97.92 | 97.20 | 97.34 | 97.34 | 96.32 | 96.77 | 97.65 | 96.81 | 96.97 | 97.99 | 97.26 | 97.38
TABLE III: Performance comparison of the SNNs trained with Q-STDB with state-of-the-art deep ANNs on IP and PU datasets Authors | ANN/SNN | Architecture |  OA ($\%$) |  AA ($\%$) |  Kappa ($\%$)
---|---|---|---|---|---
Dataset : Indian Pines
Alipour-Fard | ANN | MSKNet | 81.73 | 71.4 | 79.2
et al. (2020) [21] |  |  |  |  |
Song et al. | ANN | DFFN | 98.52 | 97.69 | 98.32
(2018)[22] |  |  |  |  |
Zhong et al. | ANN | SSRN | 99.19 | 98.93 | 99.07
(2018) [57] |  |  |  |  |
Roy et al. | ANN | HybridSN | 98.39 | 98.01 | 98.16
(2020) [25] |  |  |  |  |
Hamida et al. | ANN | 6-layer | 98.29 | 97.52 | 97.72
(2018) [23] | SNN | 3D CNN | 95.88 | 94.26 | 95.34
Luo et al. | ANN | Hybrid | 96.15 | 94.96 | 95.73
(2018) [26] | SNN | CNN | 94.90 | 94.08 | 94.78
This work | ANN | CNN-3D | 98.86 | 98.42 | 98.55
| SNN |  | 98.68 | 98.34 | 98.20
This work | ANN | CNN-32H | 97.60 | 97.08 | 97.44
| SNN |  | 97.45 | 96.73 | 96.89
Dataset : Pavia University
Alipour-Fard | ANN | MSKNet | 90.66 | 88.09 | 87.64
et al. (2020) [21] |  |  |  |  |
Song et al. | ANN | DFFN | 98.73 | 97.24 | 98.31
(2018)[22] |  |  |  |  |
Zhong et al. | ANN | SSRN | 99.61 | 99.56 | 99.33
(2018) [57] |  |  |  |  |
Hamida et al. | ANN | 6-layer | 99.32 | 99.02 | 99.09
(2018) [23] | SNN | 3D CNN | 98.55 | 98.02 | 98.28
Luo et al. | ANN | Hybrid | 99.05 | 98.35 | 98.80
(2018) [26] | SNN | CNN | 98.40 | 97.66 | 98.21
This work | ANN | CNN-3D | 99.69 | 99.42 | 99.58
| SNN |  | 99.50 | 99.18 | 99.33
This work | ANN | CNN-32H | 99.50 | 99.09 | 99.30
| SNN |  | 99.35 | 98.88 | 98.95

###  V-C ANN & SNN Inference Results

We have used the Overall Accuracy (OA), Average Accuracy (AA), and Kappa
Coefficient evaluation measures to evaluate the HSI classification performance
for our proposed architectures, similar to [23]. Here, OA represents the
number of correctly classified samples out of the total test samples. AA
represents the average of class-wise classification accuracies, and Kappa is a
statistical metric used to assess the mutual agreement between the ground
truth map and classification map. Column-$2$ in Table II shows the ANN
accuracies, column-$3$ shows the accuracy after ANN-SNN conversion with $50$
timesteps. Column-$4$ shows the accuracy when we perform our proposed training
without quantization, while columns 5 to 7 shows the SNN test accuracies
obtained with Q-STDB for different bit precisions (4 to 6 bits) of the
weights. We observe that for all the datasets, SNNs trained with 6-bit weights
result in $5.33\times$ reduction in bit-precision compared to full-precision
(32-bit) models and perform almost at par with the full precision ANNs on both
the architectures. 4-bit weights do not incur significant accuracy drop as
well, and can be used for applications demanding high energy-efficiency and
low latency. Fig. 5 shows the confusion matrix for the HSI classification
performance of the ANN and proposed SNN over the IP dataset for both the
architectures. Although the membrane potentials do not need to be quantized as
described in Section III, we observed that the model accuracy does not drop
significantly even if we quantize them, and hence, the SNN results shown in
Table II correspond to 6-bit membrane potentials. Moreover, quantized membrane
potentials can reduce the data movement cost as discussed in Section VI-B.

The performance of our ANNs and SNNs trained via Q-STDB are compared with the
current state-of-the-art ANNs used for HSI classification in Table III. Note
that mere porting the ANN architectures used in [23, 26] to SNNs, and
performing 6-bit Q-STDB results in significant accuracy drop, and hence, shows
the efficacy of our proposed architectures.

####  V-C1 Q-STDB vs Post-Training Quantization (PTQ)

PTQ cannot always yield ultra low-precision SNNs with SOTA test accuracy. For
example, for the IP dataset and CNN-32H architecture, the lowest bit precision
of the weights that the SNNs can be trained with PTQ for no more than $1\%$
reduction in SOTA test accuracy is $12$, if we limit the total number of time
steps to $5$. Fig. 4(b) shows the test accuracies for different bit precisions
(${\leq}{12}$) of weights with PTQ on IP dataset. The weights can be further
quantized to $8$-bits if we increase the time steps to $10$, which increases
the latency. On the other hand, Q-STDB results in accurate (${\leq}{1}\%$
deviation from ANN accuracy) $6$ bit SNNs with only $5$ time steps, which
improves both the energy-efficiency and latency. The energy-efficiency of our
proposed architectures trained with Q-STDB are quantified in Section VI.

####  V-C2 Affine vs Scale Quantization during Training

As illustrated in Section III, performing scale quantization during the
forward path in training degrades the SNN accuracy significantly. Fig. 4(a)
shows the test accuracies for affine and scale quantization during training
with CNN-3D architecture on IP dataset.

Figure 5: Confusion Matrix for HSI test performance of ANN and proposed 6-bit
SNN over IP dataset for both CNN-3D and CNN-32H. The ANN and SNN confusion
matrices look similar for both the network architectures. CNN-32H incurs a
little drop in accuracy compared to CNN-3D due to shallow architecture.

##  VI Improvement in Energy-Delay Product

###  VI-A Spiking Activity

To model energy consumption, we assume a generated SNN spike consumes a fixed
amount of energy [43]. Based on this assumption, earlier works [42, 31] have
adopted the average spiking activity (also known as average spike count) of an
SNN layer $l$, denoted ${\zeta}^{l}$, as a measure of compute-energy of the
model. In particular, ${\zeta}^{l}$ is computed as the ratio of the total
spike count in $T$ steps over all the neurons of the layer $l$ to the total
number of neurons in that layer. Thus lower the spiking activity, the better
the energy efficiency.

Fig. 6 shows the average number of spikes for each layer with Q-STDB when
evaluated for $200$ samples from the IP testset for the CNN-3D architecture.
Let the average be denoted by $\zeta_{l}$ which is computed by summing all the
spikes in a layer over $5$ time steps and dividing by the number of neurons in
that layer. For example, the average spike count of the $3^{rd}$ convolutional
layer of the SNN is $0.568$, which implies that over a $5$ timestep period
each neuron in that layer spikes $0.568$ times on average over all input
samples.

Figure 6: Layerwise spiking activity plots for CNN-3D on Indian Pines, Salinas
Scene and Pavia University datasets.

###  VI-B Floating point operations count (FLOPs) & Total Energy

Let us assume a 3-D convolutional layer $l$ having weight tensor
$\textbf{W}^{l}\in\mathbb{R}^{k_{l}^{x}\times k_{l}^{y}\times k_{l}^{z}\times
C_{l}^{i}\times C_{l}^{o}}$ that operates on an input activation tensor
$\textbf{I}^{l}\in\mathbb{R}^{H_{l}^{i}\times W_{l}^{i}\times C_{l}^{i}\times
D_{l}^{i}}$, where the notations are similar to the one used in Section III.
We now quantify the energy consumed to produce the corresponding output
activation tensor $\textbf{O}^{l}\in\mathbb{R}^{H_{l}^{o}\times
W_{l}^{o}\times C_{l}^{o}\times D_{l}^{o}}$ for an ANN and SNN, respectively.
Our model can be extended to fully-connected layers with $f_{l}^{i}$ and
$f_{l}^{o}$ as the number of input and output features respectively, and to
2-D convolutional layers, by shrinking a dimension of the feature maps. In
particular, for an ANN, the total number of FLOPS for layer $l$, denoted
$F_{l}^{ANN}$, is shown in row $1$ of Table IV [58, 59]. The formula can be
easily adjusted for an SNN in which the number of FLOPs at layer $l$ is a
function of the average spiking activity at the layer $(\zeta_{l})$ denoted as
$F_{l}^{SNN}$ in Table IV. Thus, as the activation output gets sparser, the
compute energy decreases.

For ANNs, FLOPs primary consist of multiply accumulate (MAC) operations of the
convolutional and linear layers. On the contrary, for SNNs, except the first
and last layer, the FLOPs are limited to accumulates (ACs) as the spikes are
binary and thus simply indicate which weights need to be accumulated at the
post-synaptic neurons. For the first layer, we need to use MAC units as we
consume analog input111Note that for the hybrid coded data input we need to
perform MAC at the first layer at $t=1$, and AC operation during remaining
timesteps at that layer. For the direct coded input, only MAC during the
$1^{st}$ timestep is sufficient, as neither the inputs nor the weights change
during remaining timesteps (i.e. $5\geq t\geq 2$). (at timestep one). Hence,
the compute energy for an ANN $(E^{ANN})$ and an iso-architecture SNN model
$(E^{SNN})$ can be written as

$\displaystyle E^{ANN}$ $\displaystyle=(\sum_{l=1}^{L}F^{SNN}_{l}){E_{MAC}}$
(15) $\displaystyle E^{SNN}$
$\displaystyle=(F^{ANN}_{1}){E_{MAC}}+(\sum_{l=2}^{L}F^{SNN}_{l}){E_{AC}}$
(16)

where $L$ is the total number of layers. Note that $E_{MAC}$ and $E_{AC}$ are
the energy consumption for a MAC and AC operation respectively. As shown in
Table V, $E_{AC}$ is $\mathord{\sim}32\times$ lower than $E_{MAC}$ [60] in
$45$ nm CMOS technology for 32-bit precision. To compute $E_{MAC}$ and
$E_{AC}$ for any arbitrary bit precision $Q$ (6-bits in our work), we use
$E_{MAC}{\propto}Q^{1.25}$ [61], and $E_{AC}{\propto}Q$ [62]. These numbers
may vary for different technologies, but generally, in most technologies, an
AC operation is significantly less expensive than a MAC operation and its’
energy scales close to linearly with bit precision.

TABLE IV: Floating point operations count (FLOPs) in Convolutional and Fully-connected layer for ANN and SNN models Model | Number of FLOPs
---|---
| Notation |  3-D Conv. layer $l$ |  2-D Conv. layer $l$ |  FC layer $l$
$ANN$ | $F_{l}^{A}NN$ | $k_{l}^{x}\times k_{l}^{y}\times k_{l}^{z}\times H_{l}^{o}\times$ | $(k_{l})^{2}\times H_{l}^{o}\times W_{l}^{o}\times$ | $f_{l}^{i}\times f_{l}^{o}$
|  | $W_{l}^{o}\times D_{l}^{o}\times C_{l}^{o}\times C_{l}^{i}$ | $C_{l}^{o}\times C_{l}^{i}$ |
$SNN$ | $F_{l}^{S}NN$ | $k_{l}^{x}\times k_{l}^{y}\times k_{l}^{z}\times H_{l}^{o}\times$ | $(k_{l})^{2}\times H_{l}^{o}\times W_{l}^{o}\times$ | $f_{l}^{i}\times f_{l}^{o}\times\zeta_{l}$
|  | $W_{l}^{o}\times D_{l}^{o}\times C_{l}^{o}\times C_{l}^{i}\times\zeta_{l}$ | $C_{l}^{o}\times C_{l}^{i}\times\zeta_{l}$ |

Fig. 7 illustrates the compute energy consumption and FLOPs for full precision
ANN and 6-bit quantized SNN models of the two proposed architectures while
classifying the IP, PU, and SS datasets, where the energy is normalized to
that of an equivalent ANN. We also consider $6$-bit ANN models to compare the
energy-efficiency of low-precision ANNs and SNNs. As seen in Fig. 7, 6-bit ANN
models are $12.5\times$ energy-efficient compared to 32-bit ANN models due to
the similar factor of improvement in MAC energy (see Table V). Note that we
can achieve the HSI test accuracies shown in Table II with quantized ANNs as
well.

The FLOPs for SNNs obtained by our proposed training framework is smaller than
that for an ANN with similar number of parameters due to low spiking activity.
Moreover, because the ACs consume significantly less energy than MACs for all
bit precisions, SNNs are significantly more compute efficient. In particular,
for CNN-3D on IP, our proposed SNN consumes $\mathord{\sim}199.3\times$ and
$\mathord{\sim}15.9\times$ less compute energy than an iso-architecture full-
precision and 6-bit ANN with similar parameters respectively. The improvements
become ${\sim}560.6\times$ and ${\sim}44.8\times$ respectively on averaging
across the two network architectures and three datasets. Note that we did not
consider the memory access energy in our evaluation because it is dependent on
the underlying system architecture. In general, SNNs incur significant data
movement because both the membrane potentials and weights need to be fetched
from the on-chip memory. Q-STDB addresses the memory cost by reducing their
bit precisions by $5.33\times$ (see Section V-C) compared to full-precision
models. Moreover, there have been many proposals to reduce the memory cost by
data buffering [63], computing in non-volatile crossbar memory arrays [64],
and data reuse with energy-efficient dataflows [65]. All these techniques can
be complemented with Q-STDB to further decrease the memory cost.

TABLE V: Estimated energy costs for $32$ and $6$-bit MAC and AC operations in 45 $nm$ CMOS process Bit-precision | Operation |  Energy ($pJ$)
---|---|---
32 | Mutiply-and-Accumulate (MAC) | $3.2$
| Accumulate (AC) | $0.1$
6 | Mutiply-and-Accumulate (MAC) | $0.26$
| Accumulate (AC) | $0.02$
Figure 7: Comparison of FLOPs and compute energy of CNN-3D and CNN-32H between
ANN and SNN models while classifying on (a) Indian Pines, (b) Salinas Scene
and (c) Pavia University datasets, respectively.

##  VII Conclusions and Broader Impact

In this paper, we propose a spiking version of a 3-D and hybrid combination of
3-D and 2-D convolutional architectures for HSI classification. We present a
quantization-aware training technique, that yields highly accurate low-
precision SNNs, which can be accelerated by integer math units or PIM
accelerators. Our quantized SNNs offer significant improvements in energy
consumption compared to both full and low-precision ANNs for HSI
classification.

Our proposal results in energy-efficient SNN models, which can be readily
deployed in HSI sensors, thereby eliminating the bandwidth and privacy
concerns of going to the cloud. Since the commercial applications of HSI
analysis are broadly expanding and the models required to train HSI are
becoming deeper, energy-efficiency becomes a key concern, as seen in
traditional computer vision tasks. To the best of our knowledge, this work is
the first to address energy-efficiency of HSI models, and can hopefully
inspire more research in low power algorithm-hardware co-design of neural
network models for HSI classification.

## References

  * [1] Y. Chen, Z. Lin, X. Zhao, G. Wang, and Y. Gu, “Deep learning-based classification of hyperspectral data,” _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , vol. 7, no. 6, pp. 2094–2107, 2014.
  * [2] Y. Wan, Y. Fan, and M. Jin, “Application of hyperspectral remote sensing for supplementary investigation of polymetallic deposits in huaniushan ore region, northwestern china,” _Scientific Reports_ , vol. 11, p. 440, 01 2021\.
  * [3] A. Papp, J. Pegoraro, D. Bauer, P. Taupe, C. Wiesmeyr, and A. Kriechbaum-Zabini, “Automatic annotation of hyperspectral images and spectral signal classification of people and vehicles in areas of dense vegetation with deep learning,” _Remote Sensing_ , vol. 12, no. 13, 2020\.
  * [4] Z. Zheng, Y. Zhong, A. Ma, and L. Zhang, “FPGA: Fast patch-free global learning framework for fully end-to-end hyperspectral image classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 58, no. 8, pp. 5612–5626, 2020.
  * [5] F. Melgani and L. Bruzzone, “Classification of hyperspectral remote sensing images with support vector machines,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 42, no. 8, pp. 1778–1790, 2004.
  * [6] M. Pal, “Random forests for land cover classification,” in _IGARSS 2003\. 2003 IEEE International Geoscience and Remote Sensing Symposium. Proceedings (IEEE Cat. No.03CH37477)_ , vol. 6, no. 1, 2003, pp. 3510–3512 vol.6.
  * [7] J. Xia, N. Yokoya, and A. Iwasaki, “Hyperspectral image classification with canonical correlation forests,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 55, no. 1, pp. 421–431, 2017.
  * [8] B. Krishnapuram, L. Carin, M. A. T. Figueiredo, and A. J. Hartemink, “Sparse multinomial logistic regression: fast algorithms and generalization bounds,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 27, no. 6, pp. 957–968, 2005.
  * [9] G. Camps-Valls, L. Gomez-Chova, J. Munoz-Mari, J. Vila-Frances, and J. Calpe-Maravilla, “Composite kernels for hyperspectral image classification,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 3, no. 1, pp. 93–97, 2006.
  * [10] B. Tu, X. Zhang, X. Kang, G. Zhang, J. Wang, and J. Wu, “Hyperspectral image classification via fusing correlation coefficient and joint sparse representation,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 15, no. 3, pp. 340–344, 2018.
  * [11] P. Gao, J. Wang, H. Zhang, and Z. Li, “Boltzmann entropy-based unsupervised band selection for hyperspectral image classification,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 16, no. 3, pp. 462–466, 2019.
  * [12] Q. Gao, S. Lim, and X. Jia, “Hyperspectral image classification using joint sparse model and discontinuity preserving relaxation,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 15, no. 1, pp. 78–82, 2018.
  * [13] J. A. Benediktsson, J. A. Palmason, and J. R. Sveinsson, “Classification of hyperspectral data from urban areas based on extended morphological profiles,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 43, no. 3, pp. 480–491, 2005.
  * [14] J. Li, P. R. Marpu, A. Plaza, J. M. Bioucas-Dias, and J. A. Benediktsson, “Generalized composite kernel framework for hyperspectral image classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 51, no. 9, pp. 4816–4829, 2013.
  * [15] A. Krizhevsky _et al._ , “ImageNet classification with deep convolutional neural networks,” in _Advances in Neural Information Processing Systems_ , 2012, pp. 1097–1105.
  * [16] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [17] S. Ren, K. He, R. Girshick, and J. Sun, “Faster R-CNN: Towards real-time object detection with region proposal networks,” _IEEE Trans. Pattern Anal. Mach. Intell._ , vol. 39, no. 6, p. 1137–1149, Jun. 2017.
  * [18] K. He, G. Gkioxari, P. Dollár, and R. Girshick, “Mask R-CNN,” _arXiv preprint arXiv:1703.06870_ , 2018.
  * [19] V. K. Repala and S. R. Dubey, “Dual CNN models for unsupervised monocular depth estimation,” _arXiv preprint arXiv:1804.06324_ , 2019.
  * [20] K. Makantasis, K. Karantzalos, A. Doulamis, and N. Doulamis, “Deep supervised learning for hyperspectral data classification through convolutional neural networks,” in _2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS)_ , vol. 1, no. 1, 2015, pp. 4959–4962.
  * [21] T. Alipour-Fard, M. E. Paoletti, J. M. Haut, H. Arefi, J. Plaza, and A. Plaza, “Multibranch selective kernel networks for hyperspectral image classification,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 1, no. 1, pp. 1–5, 2020.
  * [22] W. Song, S. Li, L. Fang, and T. Lu, “Hyperspectral image classification with deep feature fusion network,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 56, no. 6, pp. 3173–3184, 2018.
  * [23] A. Ben Hamida, A. Benoit, P. Lambert, and C. Ben Amar, “3-D deep learning approach for remote sensing image classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 56, no. 8, pp. 4420–4434, 2018.
  * [24] H. Lee and H. Kwon, “Going deeper with contextual cnn for hyperspectral image classification,” _IEEE Transactions on Image Processing_ , vol. 26, no. 10, pp. 4843–4855, 2017.
  * [25] S. K. Roy, G. Krishna, S. R. Dubey, and B. B. Chaudhuri, “HybridSN: Exploring 3-D–2-D CNN feature hierarchy for hyperspectral image classification,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 17, no. 2, pp. 277–281, 2020.
  * [26] Y. Luo, J. Zou, C. Yao, X. Zhao, T. Li, and G. Bai, “HSI-CNN: A novel convolution neural network for hyperspectral image,” in _2018 International Conference on Audio, Language and Image Processing (ICALIP)_ , vol. 1, no. 1, 2018, pp. 464–469.
  * [27] D. Li, X. Chen, M. Becchi, and Z. Zong, “Evaluating the energy efficiency of deep convolutional neural networks on CPUs and GPUs,” in _2016 IEEE International Conferences on Big Data and Cloud Computing (BDCloud), Social Computing and Networking (SocialCom), Sustainable Computing and Communications (SustainCom) (BDCloud-SocialCom-SustainCom)_ , vol. 1, no. 1, 2016, pp. 477–484.
  * [28] Hien Van Nguyen, A. Banerjee, and R. Chellappa, “Tracking via object reflectance using a hyperspectral video camera,” in _2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Workshops_ , vol. 1, no. 1, 2010, pp. 44–51.
  * [29] M. Pfeiffer and T. Pfeil, “Deep learning with spiking neurons: Opportunities and challenges,” _Frontiers in Neuroscience_ , vol. 12, p. 774, 2018.
  * [30] P. U. Diehl, G. Zarrella, A. Cassidy, B. U. Pedroni, and E. Neftci, “Conversion of artificial recurrent neural networks to spiking neural networks for low-power neuromorphic hardware,” in _2016 IEEE International Conference on Rebooting Computing (ICRC)_. IEEE, 2016, pp. 1–8.
  * [31] A. Sengupta, Y. Ye, R. Wang, C. Liu, and K. Roy, “Going deeper in spiking neural networks: VGG and residual architectures,” _Frontiers in Neuroscience_ , vol. 13, p. 95, 2019.
  * [32] N. Rathi and K. Roy, “DIET-SNN: Direct input encoding with leakage and threshold optimization in deep spiking neural networks,” _arXiv preprint arXiv:2008.03658_ , 2020.
  * [33] I. M. Comsa, K. Potempa, L. Versari, T. Fischbacher, A. Gesmundo, and J. Alakuijala, “Temporal coding in spiking neural networks with alpha synaptic function,” in _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , vol. 1, no. 1, 2020, pp. 8529–8533.
  * [34] S. R. Kheradpisheh and T. Masquelier, “Temporal backpropagation for spiking neural networks with one spike per neuron,” _International Journal of Neural Systems_ , vol. 30, no. 06, May 2020.
  * [35] J. Kim, H. Kim, S. Huh, J. Lee, and K. Choi, “Deep neural networks with weighted spikes,” _Neurocomputing_ , vol. 311, pp. 373–386, 2018.
  * [36] D. A. Almomani, M. Alauthman, M. Alweshah, O. Dorgham, and F. Albalas, “A comparative study on spiking neural network encoding schema: implemented with cloud computing,” _Cluster Computing_ , vol. 22, 06 2019.
  * [37] G. Datta, S. Kundu, and P. A. Beerel, “Training energy-efficient deep spiking neural networks with single-spike hybrid input encoding,” _arXiv preprint arXiv:2107.12374_ , 2021.
  * [38] J. H. Lee, T. Delbruck, and M. Pfeiffer, “Training deep spiking neural networks using backpropagation,” _Frontiers in Neuroscience_ , vol. 10, p. 508, 2016.
  * [39] Y. Wu, L. Deng, G. Li, J. Zhu, Y. Xie, and L. Shi, “Direct training for spiking neural networks: Faster, larger, better,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, 2019, pp. 1311–1318.
  * [40] S. Lu and A. Sengupta, “Exploring the connection between binary and spiking neural networks,” _arXiv preprint arXiv:2002.10064_ , 2020.
  * [41] C. Lee, S. S. Sarwar, P. Panda, G. Srinivasan, and K. Roy, “Enabling spike-based backpropagation for training deep neural network architectures,” _Frontiers in Neuroscience_ , vol. 14, p. 119, 2020.
  * [42] N. Rathi, G. Srinivasan, P. Panda, and K. Roy, “Enabling deep spiking neural networks with hybrid conversion and spike timing dependent backpropagation,” _arXiv preprint arXiv:2005.01807_ , 2020.
  * [43] Y. Cao, Y. Chen, and D. Khosla, “Spiking deep convolutional neural networks for energy-efficient object recognition,” _International Journal of Computer Vision_ , vol. 113, pp. 54–66, 05 2015.
  * [44] B. Rueckauer, I.-A. Lungu, Y. Hu, M. Pfeiffer, and S.-C. Liu, “Conversion of continuous-valued deep networks to efficient event-driven networks for image classification,” _Frontiers in Neuroscience_ , vol. 11, p. 682, 2017.
  * [45] P. U. Diehl, D. Neil, J. Binas, M. Cook, S. Liu, and M. Pfeiffer, “Fast-classifying, high-accuracy spiking deep networks through weight and threshold balancing,” in _2015 International Joint Conference on Neural Networks (IJCNN)_ , vol. 1, no. 1, 2015, pp. 1–8.
  * [46] Y. Hu, H. Tang, and G. Pan, “Spiking deep residual network,” _arXiv preprint arXiv:1805.01352_ , 2018.
  * [47] P. Panda and K. Roy, “Unsupervised regenerative learning of hierarchical features in spiking deep networks for object recognition,” _arXiv preprint arXiv:1602.01510_ , 2016.
  * [48] G. Bellec, D. Salaj, A. Subramoney, R. Legenstein, and W. Maass, “Long short-term memory and learning-to-learn in networks of spiking neurons,” _arXiv preprint arXiv:1803.09574_ , 2018.
  * [49] E. O. Neftci, H. Mostafa, and F. Zenke, “Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks,” _IEEE Signal Processing Magazine_ , vol. 36, no. 6, pp. 51–63, 2019.
  * [50] S. R. Jain, A. Gural, M. Wu, and C. H. Dick, “Trained quantization thresholds for accurate and efficient fixed-point inference of deep neural networks,” _arXiv preprint arXiv:1903.08066_ , 2020.
  * [51] N. Rathi, P. Panda, and K. Roy, “STDP based pruning of connections and weight quantization in spiking neural networks for energy efficient recognition,” _arXiv preprint arXiv:1710.04734_ , 2017.
  * [52] M. B. G. Sulaiman, K. C. Juang, and C. C. Lu, “Weight quantization in spiking neural network for hardware implementation,” in _2020 IEEE International Conference on Consumer Electronics - Taiwan (ICCE-Taiwan)_ , vol. 1, no. 1, 2020, pp. 1–2.
  * [53] M. Courbariaux, I. Hubara, D. Soudry, R. El-Yaniv, and Y. Bengio, “Binarized neural networks: Training deep neural networks with weights and activations constrained to +1 or -1,” _arXiv preprint arXiv:1602.02830_ , 2016.
  * [54] N. Bjorck, C. P. Gomes, B. Selman, and K. Q. Weinberger, “Understanding batch normalization,” in _Advances in Neural Information Processing Systems_ , 2018, pp. 7694–7705.
  * [55] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A simple way to prevent neural networks from overfitting,” _Journal of Machine Learning Research_ , vol. 15, pp. 1929–1958, 06 2014\.
  * [56] M. Graña, M. A. Veganzons, and B. Ayerdi, “Hyperspectral remote sensing scenes,” http://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes.
  * [57] Z. Zhong, J. Li, Z. Luo, and M. Chapman, “Spectral–spatial residual network for hyperspectral image classification: A 3-D deep learning framework,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 56, no. 2, pp. 847–858, 2018.
  * [58] S. Kundu, M. Nazemi, M. Pedram, K. M. Chugg, and P. Beerel, “Pre-defined sparsity for low-complexity convolutional neural networks,” _IEEE Transactions on Computers_ , 2020.
  * [59] S. Kundu, S. Prakash, H. Akrami, P. A. Beerel, and K. M. Chugg, “pSConv: A pre-defined sparse kernel based convolution for deep CNNs,” in _2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)_. IEEE, 2019, pp. 100–107.
  * [60] M. Horowitz, “1.1 Computing’s energy problem (and what we can do about it),” in _2014 IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC)_. IEEE, 2014, pp. 10–14.
  * [61] B. Moons, R. Uytterhoeven, W. Dehaene, and M. Verhelst, “14.5 envision: A 0.26-to-10TOPS/W subword-parallel dynamic-voltage-accuracy-frequency-scalable convolutional neural network processor in 28nm fdsoi,” in _2017 IEEE International Solid-State Circuits Conference (ISSCC)_ , vol. 1, no. 1, 2017, pp. 246–247.
  * [62] W. Simon, J. Galicia, A. Levisse, M. Zapater, and D. Atienza, “A fast, reliable and wide-voltage-range in-memory computing architecture,” in _2019 56th ACM/IEEE Design Automation Conference (DAC)_ , vol. 1, no. 1, 2019, pp. 1–6.
  * [63] Y. Shen, M. Ferdman, and P. Milder, “Escher: A CNN accelerator with flexible buffering to minimize off-chip transfer,” in _2017 IEEE 25th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM)_ , vol. 1, no. 1, 2017, pp. 93–100.
  * [64] B. Chen, F. Cai, J. Zhou, W. Ma, P. Sheridan, and W. D. Lu, “Efficient in-memory computing architecture based on crossbar arrays,” in _2015 IEEE International Electron Devices Meeting (IEDM)_ , vol. 1, no. 1, 2015, pp. 1–4.
  * [65] Y.-H. Chen, J. Emer, and V. Sze, “Eyeriss: A spatial architecture for energy-efficient dataflow for convolutional neural networks,” _ACM SIGARCH Computer Architecture News_ , vol. 44, 06 2016.

|  Gourav Datta received his bachelors’ degree in Instrumentation Engineering
with a minor in Electronics and Electrical Communication Engineering from
Indian Institute of Technology (IIT) Kharagpur, India in 2018. He then joined
the Ming Hsieh Department of Electrical and Computer Engineering at the
University of Southern California where he is currently pursuing a PhD degree.
He interned at Apple Inc. and INRIA Research Centre in the summers of 2019 and
2017, respectively. His research focuses on the entire computing stack,
including devices, circuits, architectures and algorithms for accelerating
machine learning workloads. During his tenure at IIT Kharagpur, he has
received the Institute Silver medal and was adjudged the best outgoing student
in academics in his batch.
---|---
|  Souvik Kundu received his M. Tech degree in Microelectronics and VLSI
design from Indian Institute of Technology Kharagpur, India in 2015. He worked
as R & D Engineer II at Synopsys India Pvt. Ltd. and as Digital Design
Engineer at Texas Instruments India Pvt. Ltd. from 2015 to 2016 and from 2016
to 2017, respectively. He is currently working towards the Ph.D. degree in
Electrical and Computer Engineering at the University of Southern California,
Los Angeles, CA, USA. His research focuses on energy aware sparsity, model
search, algorithm-hardware co-design of robust and energy-efficient neural
networks for CMOS and beyond CMOS technology.
---|---
|  Akhilesh R. Jaiswal is a Research Assistant Professor of Electrical and
Computer Engineering and a Scientist at USC’s Information Sciences Institute’s
(ISI) Application Specific Intelligent Computing (ASIC) Lab. Prior to USC/ISI,
Dr. Jaiswal was a Senior Research Engineer with GLOBALFOUNDIRES (GF) at Malta.
Dr. Jaiswal received his Ph.D. degree in Nano-electronics from Purdue
University in May 2019. As a part of doctoral program his research focused on
1) CMOS based analog and digital in-memory and near-memory computing using
standard memory bit-cells for beyond von-Neumann computing. 2) Exploration of
bio-mimetic devices and circuits using emerging non-volatile technologies for
Neuromorphic computing. His current research interest includes exploration of
’alternate computing paradigms’ using ’alternate state variables’. Dr. Jaiswal
has authored several publications and holds 15+ issued and several pending
patents with the USPTO.
---|---
|  Peter A. Beerel received his B.S.E. degree in Electrical Engineering from
Princeton University, Princeton, NJ, in 1989 and his M.S. and Ph.D. degrees in
Electrical Engineering from Stanford University, Stanford, CA, in 1991 and
1994, respectively. He then joined the Ming Hsieh Department of Electrical and
Computer Engineering at the University of Southern California where he is
currently a professor and the Associate Chair of the Computer Engineering
Division. He is also a Research Director at the Information Science Institute
at USC. Previously, he co-founded TimeLess Design Automation to commercialize
an asynchronous ASIC flow in 2008 and sold the company in 2010 to Fulcrum
Microsystems which was bought by Intel in 2011. His interests include a
variety of topics in computer-aided design, machine learning, hardware
security, and asynchronous VLSI and the commercialization of these
technologies. He is a Senior Member of the IEEE.
---|---


# Notes on the norm of pre-Schwarzian derivatives of certain analytic
functions

R. Kargar  Young Researchers and Elite Club, Ardabil Branch, Islamic Azad
University, Ardabil, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we obtain sharp bounds for the norm of pre–Schwarzian
derivatives of certain analytic functions. Initially this problem was handled
by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud Univ Babeş–Bolyai Math
61(2): 155–162, 2016]. We pointed out that the proofs by Rahmatan et al. are
incorrect and present correct proofs.

###### Key words and phrases:

Analytic; Univalent; Locally univalent; Subordination; Pre–Schwarzian norm.

###### 2010 Mathematics Subject Classification:

30C45

##  1\. Introduction

Let $\Delta$ be the open unit disc on the complex plane $\mathbb{C}$. Let
$\mathcal{H}$ be the family of all analytic functions and
$\mathcal{A}\subset\mathcal{H}$ be the family of all normalized functions in
$\Delta$. We denote by $\mathcal{U}$ the class of all univalent functions in
$\Delta$ and denote by $\mathcal{LU}\subset\mathcal{H}$ the class of all
locally univalent functions in $\Delta$. For a $f\in\mathcal{LU}$, we consider
the following norm

$||f||=\sup_{z\in\Delta}(1-|z|^{2})\left|\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right|,$

where the quantity $f^{\prime\prime}/f$ is often referred to as pre–Schwarzian
derivative of $f$ and in the theory of Teichmüller spaces is considered as
element of complex Banach spaces. We remark that $||f||<\infty$ if, and only
if, $f$ is uniformly locally univalent in $\Delta$. Note that, $||f||\leq 6$
if $f$ is univalent in $\Delta$ and, conversely, $f$ is univalent in $\Delta$
if $||f|\leq 1$. Both of these bounds are sharp, see [1]. For more geometric
properties of the function $f$ relating the norm, see [2, 4, 7] and the
references therein.

We say that a function $f$ is subordinate to $g$, written by $f(z)\prec g(z)$
or $f\prec g$ where $f$ and $g$ belonging to the class $\mathcal{A}$, if there
exists a Schwarz function $w(z)$ is analytic in $\Delta$ with

$w(0)=0\quad{\rm and}\quad|w(z)|<1\quad(z\in\Delta),$

such that $f(z)=g(w(z))$ for all $z\in\Delta$.

In the sequel, we recall two definition which are certain subclasses of
analytic and normalized functions $\mathcal{A}$. First, we say that a function
$f\in\mathcal{A}$ belongs to the class $\mathcal{S}(\alpha,\beta)$ if it
satisfies the following two–sided inequality

$\alpha<{\rm
Re}\left\\{\frac{zf^{\prime}(z)}{f(z)}\right\\}<\beta\quad(z\in\Delta),$

where $0\leq\alpha<1$ and $\beta>1$. The class $\mathcal{S}(\alpha,\beta)$ was
introduced by Kuroki and Owa (cf. [6]). Also, we say that a function
$f\in\mathcal{A}$ belongs to the class $\mathcal{V}(\alpha,\beta)$ if

$\alpha<{\rm
Re}\left\\{\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)\right\\}<\beta\quad(z\in\Delta).$

The class $\mathcal{V}(\alpha,\beta)$ was first introduced by Kargar et al.,
see [5].

Since the convex univalent function

(1.1)
$P_{\alpha,\beta}(z)=1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}z}{1-z}\right)\quad(z\in\Delta),$

where

(1.2) $\phi:=\frac{2\pi(1-\alpha)}{\beta-\alpha},$

maps $\Delta$ onto the domain $\Omega=\\{\omega:\alpha<{\rm
Re}\\{\omega\\}<\beta\\}$ conformally, thus we have.

######  Lemma 1.1.

([6, Lemma 1.3]) Let $0\leq\alpha<1$ and $\beta>1$. Then
$f\in\mathcal{S}(\alpha,\beta)$ if, and only if,

$\frac{zf^{\prime}(z)}{f(z)}\prec
1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}z}{1-z}\right)\quad(z\in\Delta),$

where $\phi$ is defined in (1.2).

######  Lemma 1.2.

([5, Lemma 1.1]) Let $0\leq\alpha<1$ and $\beta>1$. Then
$f\in\mathcal{V}(\alpha,\beta)$ if, and only if,

$\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)\prec
1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}z}{1-z}\right)\quad(z\in\Delta),$

where $\phi$ is defined in (1.2).

Rahmatan et al. (see [8]) estimated the norm of pre–Schwarzian derivatives of
the function $f$ where $f$ belong to the classes $\mathcal{S}(\alpha,\beta)$
and $\mathcal{V}(\alpha,\beta)$. Both estimates and proofs are incorrect.
Indeed, the estimates of $||f||$ were wrongly proven by Rahmatan, Najafzadeh
and Ebadian are in the following form:

Theorem A: For $0\leq\alpha<1<\beta$, if $f\in\mathcal{S}(\alpha,\beta)$, then

$||f||\leq\frac{2(\beta-\alpha)}{\pi}\left(1-e^{2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right).$

Theorem B: For $0\leq\alpha<1<\beta$, if $f\in\mathcal{V}(\alpha,\beta)$, then

$||f||\leq\frac{3(\beta-\alpha)}{\pi}\left(1-e^{2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right).$

First, note that both bounds are complex numbers.

In this paper we give the best estimate for $||f||$ when
$f\in\mathcal{S}(\alpha,\beta)$ and disprove the Theorem B. However, we show
that $||f||<\infty$ when $f\in\mathcal{V}(\alpha,\beta)$.

##  2\. The Main Results

The first result of the paper is the following.

######  Theorem 2.1.

Let $0\leq\alpha<1$ and $\beta>1$. If a function $f$ belongs to the class
$\mathcal{S}(\alpha,\beta)$, then

(2.1)
$||f||\leq\frac{2(\beta-\alpha)}{\pi}\sqrt{4\sin^{2}(\phi/2)+2\pi^{2}}-\frac{4\sin(\phi/2)}{\sqrt{4\sin^{2}(\phi/2)+2\pi^{2}}},$

where $\phi$ is defined in (1.2). The result is sharp.

###### Proof.

Let that $0\leq\alpha<1$, $\beta>1$ and $\phi$ be given by (1.2). If
$f\in\mathcal{S}(\alpha,\beta)$, by Lemma 1.1, then we have

(2.2) $\frac{zf^{\prime}(z)}{f(z)}\prec
1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}z}{1-z}\right)\quad(z\in\Delta).$

The above subordination relation (2.2) implies that

$\frac{zf^{\prime}(z)}{f(z)}=1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\quad(z\in\Delta),$

or equivalently

(2.3)
$\log\left\\{\frac{zf^{\prime}(z)}{f(z)}\right\\}=\log\left\\{1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right\\}\quad(z\in\Delta),$

where $w(z)$ is the Schwarz function. From (2.3), differentiating on both
sides, after simplification, we obtain

(2.4)
$\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}=\frac{(\beta-\alpha)i}{\pi}\left[\frac{1}{z}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)+\frac{(1-e^{i\phi})w^{\prime}(z)}{(1-w(z))(1-e^{i\phi}w(z))\left(1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right)}\right].$

It is well–known that $|w(z)|\leq|z|$ (cf. [3]) and also by the Schwarz–Pick
lemma, for a Schwarz function the following inequality

(2.5) $|w^{\prime}(z)|\leq\frac{1-|w(z)|^{2}}{1-|z|^{2}}\quad(z\in\Delta),$

holds. Also, we know that if $\log$ is the principal branch of the complex
logarithm, then we have

(2.6) $\log z=\ln|z|+i\arg z\quad(z\in\Delta\setminus\\{0\\},-\pi<\arg
z\leq\pi).$

Therefore, by the above equation (2.6), it is well–known that if $|z|\geq 1$,
then

(2.7) $|\log z|\leq\sqrt{|z-1|^{2}+\pi^{2}},$

while for $0<|z|<1$, we have

(2.8) $|\log z|\leq\sqrt{\left|\frac{z-1}{z}\right|^{2}+\pi^{2}}.$

Thus, it is natural to distinguish the following cases.
Case 1: $\left|\frac{1-e^{i\phi}w(z)}{1-w(z)}\right|\geq 1$.
By (2.7), we have

$\displaystyle\left|\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right|$
$\displaystyle\leq\sqrt{\left|\frac{1-e^{i\phi}w(z)}{1-w(z)}-1\right|^{2}+\pi^{2}}$
$\displaystyle=\frac{\sqrt{|1-e^{i\phi}|^{2}|w(z)|^{2}+\pi^{2}|1-w(z)|^{2}}}{|1-w(z)|}$
$\displaystyle\leq\frac{\sqrt{4\sin^{2}(\phi/2)|w(z)|^{2}+\pi^{2}(1+|w(z)|^{2})}}{1-|w(z)|}$
(2.9)
$\displaystyle\leq\frac{\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}}{1-|z|}$

for all $z\in\Delta$. We note that the above inequality is well defined also
for $z=0$. Thus from (2.4), (2.5) and (2), we get

$\displaystyle\quad\left|\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right|$
$\displaystyle=\left|\frac{(\beta-\alpha)i}{\pi}\left[\frac{1}{z}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)+\frac{(1-e^{i\phi})w^{\prime}(z)}{(1-w(z))(1-e^{i\phi}w(z))\left(1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right)}\right]\right|$
$\displaystyle\leq\frac{(\beta-\alpha)}{\pi}\left[\frac{1}{|z|}\left|\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right|+\frac{\left|1-e^{i\phi}\right||w^{\prime}(z)|}{\left|1-w(z)\right|\left|1-e^{i\phi}w(z)\right|\left(1-\frac{(\beta-\alpha)}{\pi}\left|\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right|\right)}\right]$
$\displaystyle\leq\frac{(\beta-\alpha)}{\pi}\left[\frac{1}{|z|}\left\\{\frac{\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}}{1-|z|}\right\\}+\frac{2\sin(\phi/2)}{1-|z|-\frac{(\beta-\alpha)}{\pi}\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}}.\frac{1+|z|}{1-|z|^{2}}\right].$

However, we obtain

$\displaystyle||f||$
$\displaystyle=\sup_{z\in\Delta}(1-|z|^{2})\left|\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right|$
$\displaystyle\leq\sup_{z\in\Delta}\left\\{\frac{(\beta-\alpha)}{\pi}\left[\frac{1+|z|}{|z|}\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}+\frac{2\sin(\phi/2)(1+|z|)}{1-|z|-\frac{(\beta-\alpha)}{\pi}\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}}\right]\right\\}$
$\displaystyle=\frac{2(\beta-\alpha)}{\pi}\sqrt{4\sin^{2}(\phi/2)+2\pi^{2}}-\frac{4\sin(\phi/2)}{\sqrt{4\sin^{2}(\phi/2)+2\pi^{2}}}$

and concluding the inequality (2.1).
Case 2: $\left|\frac{1-e^{i\phi}w(z)}{1-w(z)}\right|<1$.
By (2.8), we have

$\displaystyle\left|\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right|$
$\displaystyle\leq\sqrt{\left|\frac{\frac{1-e^{i\phi}w(z)}{1-w(z)}-1}{\frac{1-e^{i\phi}w(z)}{1-w(z)}}\right|^{2}+\pi^{2}}$
$\displaystyle=\frac{\sqrt{|1-e^{i\phi}|^{2}|w(z)|^{2}+\pi^{2}|1-e^{i\phi}w(z)|^{2}}}{|1-e^{i\phi}w(z)|}$
$\displaystyle\leq\frac{\sqrt{4\sin^{2}(\phi/2)|w(z)|^{2}+\pi^{2}(1+|w(z)|^{2})}}{1-|w(z)|}\quad(|e^{i\phi}|=1)$
$\displaystyle\leq\frac{\sqrt{4\sin^{2}(\phi/2)|z|^{2}+\pi^{2}(1+|z|^{2})}}{1-|z|}.$

Since in the Cases 1 and 2 we have the equal estimates for

$\left|\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right|,$

therefore, in this case also, the desired result will be achieved. For the
sharpness, consider the function $f_{\alpha,\beta}(z)$ as follows

$\displaystyle f_{\alpha,\beta}(z)$
$\displaystyle=z\exp\left\\{\frac{(\beta-\alpha)i}{\pi}\int_{0}^{z}\frac{1}{\xi}\log\left(\frac{1-e^{i\phi}\xi}{1-\xi}\right)d\xi\right\\}$
$\displaystyle=z+\frac{(\beta-\alpha)i}{\pi}\left(1-e^{i\phi}\right)z^{2}+\cdots,$

where $\phi$ is defined in (1.2), $0\leq\alpha<1$ and $\beta>1$. A simple
calculation, gives us

$\frac{zf^{\prime}_{\alpha,\beta}(z)}{f_{\alpha,\beta}(z)}=1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}z}{1-z}\right)\quad(z\in\Delta)$

and thus $f_{\alpha,\beta}(z)\in\mathcal{S}(\alpha,\beta)$. With the same
proof as above we get the desired result. Also, the result is sharp for a
rotation of the function $f_{\alpha,\beta}(z)$ as follows:

$\mathfrak{f}_{\alpha,\beta}(z)=z\exp\left\\{\frac{(\beta-\alpha)i}{\pi}\int_{0}^{z}\frac{1}{\xi}\log\left(\frac{1-e^{i\phi}\xi}{1-e^{-i\phi}\xi}\right)d\xi\right\\}.$

This is the end of proof. ∎

######  Remark 2.1.

In the Theorem B, the authors estimated $||f||$ when
$f\in\mathcal{V}(\alpha,\beta)$. But in the proof of this theorem [8, p. 160],
wrongly, they used from the following equation

$\frac{zf^{\prime}(z)}{f(z)}=P_{\alpha,\beta}(w(z)),$

where $P_{\alpha,\beta}$ is defined in (1.1). This means that $f$,
simultaneously, belonging to the class $\mathcal{S}(\alpha,\beta)$ and
$\mathcal{V}(\alpha,\beta)$. Next, we show that the best estimate for $||f||$
when $f\in\mathcal{V}(\alpha,\beta)$ does not exists.

######  Theorem 2.2.

Let $0\leq\alpha<1$ and $\beta>1$. If a function $f$ belongs to the class
$\mathcal{V}(\alpha,\beta)$, then $||f||<\infty$.

###### Proof.

Let $0\leq\alpha<1$, $\beta>1$ and $f\in\mathcal{V}(\alpha,\beta)$. Then by
Lemma 1.2 and by use of definition of subordination, we have

(2.10)
$\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)=P_{\alpha,\beta}(w(z))=1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\quad(z\in\Delta),$

where $w$ is Schwarz function and $\phi$ is defined in (1.2). Taking logarithm
on both sides of (2.10) and differentiating, we get

(2.11)
$\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}=2\left(\frac{f^{\prime}(z)}{f(z)}-\frac{1}{z}\right)+\frac{(\beta-\alpha)i}{\pi}\left[\frac{(1-e^{i\phi})w^{\prime}(z)}{(1-w(z))(1-e^{i\phi}w(z))\left(1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right)}\right].$

With a simple calculation, (2.10) implies that

(2.12)
$\left(\frac{f^{\prime}(z)}{f(z)}-\frac{1}{z}\right)=\frac{f(z)}{z}\left(\frac{P_{\alpha,\beta}(w(z))}{z}-1\right).$

Combining (2.11) and (2.12), give us

$\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}=2\left(\frac{f(z)}{z}\left(\frac{P_{\alpha,\beta}(w(z))}{z}-1\right)\right)+\frac{(\beta-\alpha)i}{\pi}\left[\frac{(1-e^{i\phi})w^{\prime}(z)}{(1-w(z))(1-e^{i\phi}w(z))\left(1+\frac{(\beta-\alpha)i}{\pi}\log\left(\frac{1-e^{i\phi}w(z)}{1-w(z)}\right)\right)}\right]$

It was proved in ([5, Theorem 2.2]) that if $f\in\mathcal{V}(\alpha,\beta)$
where $0<\alpha\leq 1/2$ and $\beta>1$, then

$1-\frac{1}{\alpha}<{\rm
Re}\left\\{\frac{f(z)}{z}\right\\}<\infty\quad(z\in\Delta).$

Since ${\rm Re}\\{z\\}\leq|z|$, the last two–sided inequality means that
$|f(z)/z|<\infty$ when $f\in\mathcal{V}(\alpha,\beta)$. Thus from the above we
deduce that

$\left|\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right|<\infty\quad(z\in\Delta)$

and concluding the proof. ∎

## References

  * [1] Becker, J., Pommerenke, Ch., Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354 (1984), 74–94.
  * [2] Choi, J.H., Kim, Y.C., Ponnusamy, S., Sugawa, T., Norm estimates for the Alexander transforms of convex functions of order alpha, J. Math. Anal. Appl., 303 (2005), no. 2, 661–668.
  * [3] Duren, P.L., Univalent Functions, Springer–Verlag, New York, 1983.
  * [4] Kim., Y.C., Sugawa, T., Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math., 32 (2002), no. 1, 179–200.
  * [5] Kargar, R., Ebadian, A., Sokół, J., On subordination of some analytic functions, Sib. Math. J., 57 (2016), no. 4, 599–605.
  * [6] Kuroki, K., Owa, S., Notes on new class for certain analytic functions, RIMS Kokyuroku Kyoto Univ., 1772 (2011), 21–25.
  * [7] Ponnusamy, S., Sahoo, S.K., Sugawa, T., Radius problems associated with pre–Schwarzian and Schwarzian derivatives, Analysis, 34 (2014), no. 2, 163–171.
  * [8] Rahmatan, H., Najafzadeh, Sh., Ebadian, A., The norm of pre–Schwarzian derivatives of certain analytic functions with bounded positive real part, Stud. Univ. Babeş–Bolyai Math., 61 (2016), no. 2, 155–162.