# A Matrix Ensemble Kalman Filter-based Multi-arm Neural Network to Adequately
Approximate Deep Neural Networks

Ved Piyush  Department of Statistics, University of Nebraska, Lincoln  Yuchen
Yan  Nebraska Food for Health Center, Department of Food Science and
Technology, University of Nebraska, Lincoln  Yuzhen Zhou  Amazon, Bellevue,
Washington  Yanbin Yin  Nebraska Food for Health Center, Department of Food
Science and Technology, University of Nebraska, Lincoln  Souparno Ghosh
Department of Statistics, University of Nebraska, Lincoln

###### Abstract

Deep Learners (DLs) are the state-of-art predictive mechanism with
applications in many fields requiring complex high dimensional data
processing. Although conventional DLs get trained via gradient descent with
back-propagation, Kalman Filter (KF)-based techniques that do not need
gradient computation have been developed to approximate DLs. We propose a
multi-arm extension of a KF-based DL approximator that can mimic DL when the
sample size is too small to train a multi-arm DL. The proposed Matrix Ensemble
Kalman Filter-based multi-arm ANN (MEnKF- ANN) also performs explicit model
stacking that becomes relevant when the training sample has an unequal-size
feature set. Our proposed technique can approximate Long Short-term Memory
(LSTM) Networks and attach uncertainty to the predictions obtained from these
LSTMs with desirable coverage. We demonstrate how MEnKF-ANN can “adequately”
approximate an LSTM network trained to classify what carbohydrate substrates
are digested and utilized by a microbiome sample whose genomic sequences
consist of polysaccharide utilization loci (PULs) and their encoded genes. The
scripts to reproduce the results in this paper are available at
https://github.com/Ved-Piyush/MEnKF-ANN-PUL.

##  1 Introduction

Deep Learners (DLs) have achieved state-of-art status in empirical predictive
modeling in a wide array of fields. The ability of DLs to synthesize vast
amounts of complex data, ranging from high dimensional vectors to functions
and images, to produce accurate predictions has made them the go-to models in
several areas where predictive accuracy is of paramount interest.
Bioinformatics has also seen a steep increase in articles developing or
deploying DL techniques in recent years (Min et al., 2017). However,
conventional DLs trained via gradient descent with back-propagation require
tuning of a large number of hyperparameters. Additionally, given the vast
number of weights that DLs estimate, they are prone to overfitting when the
training sample size is relatively small. Since the gradient descent
algorithms compute the weights deterministically, DLs in their vanilla form do
not yield any uncertainty estimate.

Several techniques have been proposed to alleviate the foregoing issues in
DLs. For instance, Bayesian Neural Network (BNN) (Kononenko, 1989; Neal, 2012)
was explicitly devised to incorporate epistemic and aleatoric uncertainty in
the parameter estimation process by assigning suitable priors on the weights.
The Bayesian mechanism can process these priors and generate uncertainty
associated with DL predictions. Additionally, with judicious choice of priors,
BNNs can be made less prone to overfitting (Fortuin et al., 2021; Srivastava
et al., 2014). Variational inference is another popular technique for
uncertainty quantification in DLs. In particular, Hinton and Van Camp (1993)
showed that a posterior distribution for the model weights could be obtained
by minimizing the Kullback-Leibler distance between a variational
approximation of the posterior and the true posterior of the weights. The
Bayes by Backprop (Blundell et al., 2015) is another technique that uses
variational formulation to extract uncertainty associated with the weights in
DLs.

However, the Monte Carlo dropout technique (Srivastava et al., 2014), wherein
each neuron (and all of its connections) is randomly dropped with some
probability during the model training process, has arguably turned out to be
the most popular method to regularize DLs and extract predictive uncertainty.
In addition to the conceptual simplicity of the dropout technique, it was
shown that the models trained using dropout are an approximation to Gaussian
processes and are theoretically equivalent to variational inference (Gal and
Ghahramani, 2016). Regardless of its conceptual simplicity and theoretical
underpinning, dropout methods require gradient computation . Hence, it can be
quite computationally intensive in DLs with millions of parameters.

Another suite of methods for approximating DLs uses Kalman Filters (KF), or
its variants, to obtain approximate estimates of the DL parameters (Yegenoglu
et al., 2020; Rivals and Personnaz, 1998; Wan and Van Der Merwe, 2000; Julier
and Uhlmann, 2004; Chen et al., 2019). In particular, the Ensemble Kalman
Filter (EnKF) technique offers a computationally fast approximation technique
to DLs. For instance, Chen et al. (2019) train a single hidden layer neural
network using the EnKF updating equations outlined in Iglesias et al. (2013)
and show how using the augmented state variable, one can estimate the
measurement error variance. In the DL setting, Chen et al. (2018) demonstrate
the utility of EnKF in approximating a Long Short Term Memory (LSTM) model.
Yegenoglu et al. (2020) use the EnKF to train a Convolutional Neural Network
directly using the Kalman Filtering equations derived in Iglesias et al.
(2013). All these methods approximate single-arm DLs and, therefore, cannot be
used in situations where input features can be represented in multiple ways.
Our target is to develop a multi-arm approximator to the DL. In principle, we
can train different DL approximators for each representation and perform a
post-training model averaging. However, that would increase the computation
cost substantially. We argue that since multiple different feature
representations do not necessarily offer complimentary information, developing
a multi-arm approximator that performs model averaging while training would
perform “adequately”. To that end, we develop a Matrix Ensemble Kalman Filter
(MEnKF)-based multi-arm ANN that approximates a deep learner and
simultaneously performs model averaging.

We apply our method to approximate an LSTM model trained to classify what
carbohydrate substrates are digested and utilized by a microbiome sample
characterized by genomic sequences consisting of polysaccharide utilization
loci (PULs) (Bjursell et al., 2006) and their encoded genes. We use two
different representations of the genomic sequences consisting of the PULs in
two different arms, our MEnKF-ANN approximator, and demonstrate that our
approximator closely follows the predicted probabilities obtained from the
trained LSTM. We also generate prediction intervals around the LSTM-predicted
probabilities. Our results show that the average width of the prediction
interval obtained from the MEnKF-ANN approximator is lower than that obtained
from the original LSTM trained with MC dropout. We also perform extensive
simulations, mimicking the focal dataset, to demonstrate that our method has
desirable coverage for test samples compared to the MC dropout technique.
Finally, we emphasize that even though the original problem is binary
classification, our MEnKF-ANN approximator is designed to emulate the
probabilities obtained from the original LSTM model and quantify the
uncertainties in the LSTM-predicted probabilities.

The remainder of the article is organized as follows: In section 2, we
describe the aspects of an experimentally obtained microbiome dataset that
motivated us to design this approximator. In section 3, for the sake of
completion, we offer a brief review of KF, EnKF, and how these techniques have
been used to train DLs. Section 4 details the construction of our MEnKF-ANN
method. In section 5, we offer extensive simulation results under different
scenarios and follow it up with the application on real data in section 6.
Finally, section 7 offers concluding remarks and future research directions.

##  2 Motivating Problem

The human gut, especially the colon, is a carbohydrate-rich environment
(Kaoutari et al., 2013). However, most of the non-starch polysaccharides (for
example, xylan, pectin, resistant glycans) reach the colon undegraded (Pudlo
et al., 2022) because human digestive system does not produce the enzymes
required to degrade these polysaccharides (Flint et al., 2012). Instead,
humans have developed a complex symbiotic relationship with gut microbiota,
with the latter providing a large set of enzymes for degrading the
aforementioned non-digestible dietary components (Valdes et al., 2018).
Consequently, an essential task in studying the human gut microbiome is to
predict what carbohydrate substrates a microbiome sample can digest from the
genetic characterization of the said microbiome (Koropatkin et al., 2012).

In order to generate a focused genetic characterization of the microbes that
relates to their carbohydrate utilization property, one often investigates the
genes encoding the Carbohydrate Active Enzymes (CAZymes) and other proteins
that target glycosidic linkages and act to degrade, synthesize, or modify
carbohydrates (Lombard et al., 2014; Zhang et al., 2018). This set of genes
tend to form physically linked gene clusters in the genome known as
polysaccharide utilization loci (PULs) (Bjursell et al., 2006). Consequently,
the gene sequences associated with PULs of microbes could be used as a
predictor to ascertain the carbohydrate substrate the microbe can efficiently
degrade. However, these gene sequences are string-valued quantities (Huang et
al., 2018; Stewart et al., 2018) and hence their naive quantitative
representations (for instance, one-hot-encoding or count vectorization) often
do not produce classifiers with acceptable accuracy (Badjatiya et al., 2017).
Instead, we can use LSTM to process the entire sequence of string-valued
features and then implement a classifier with a categorical loss function. The
trained LSTM, by default, produces an embedding of the gene sequences in a
vector space. Alternatively, we can also use a Doc2Vec embedding of the entire
sequence associated with the PUL or an arithmetic average of Word2Vec
embedding of each gene in the sequence and train a shallow learner (an ANN,
for example). Since various representations of the predictors are available,
we can train a multi-arm DL that takes different representations of features
in different arms and performs concatenation/late integration of the
embeddings before the prediction layer (Liu et al., 2020; Sharifi-Noghabi et
al., 2019). However, such multi-arm DLs require a relatively large number of
training samples - typically tens of thousands.

Since the experimental characterization of new PULs for carbohydrate
utilization is an expensive process (Ausland et al., 2021), we do not have
large enough labeled samples to train complex multi-arm DLs. This predicament
motivates us to develop a multi-arm approximator to a DL with the following
capabilities: (a) it must “adequately” approximate the focal single-arm DL,
(b) it should be able to ingest different feature representations in different
arms and perform model averaging, (c) it should be able to detect if the set
of representations supplied to it is substantially different from the
representations used to train the original DL, i.e., sensitive to major
misspecification. Since the original DL is trained on a single representation
of features and the approximator is ingesting multiple representations, the
latter is misspecified in a strict sense. However, this misspecification is
deliberately introduced to circumvent training a multi-arm DL and assess, via
model averaging, whether there is any benefit in using multiple
representations of the same feature set.

We extract the dataset from the dbCAN-PUL database (Ausland et al., 2021) that
contains experimentally verified PULs and the corresponding GenBank sequences
of these PULs along with known target carbohydrate substrates. Figure 1 shows
an example of a gene sequence associated with a PUL for the substrate Pectin.
We have a total of approximately 411 data points. Figure 2 shows the dataset’s
frequency distribution of various target substrates. We do not have sufficient
samples to train a complex DL to classify all the available substrates. Hence
we propose to classify the two most frequently occurring target substrates -
Xylan and Pectin - and train an LSTM binary classifier. Seventy-four samples
belong to these two classes of substrates in a reasonably balanced way. One
way to attach uncertainty to the probabilities estimated by the LSTM
architecture is to activate the dropout layers during the prediction phase.
This will generate multiple copies of the prediction depending on the
architecture of the dropout layers. However, we need to decide on how many
dropout layers to include and where to place them. Often the prediction
intervals are pretty sensitive to the number and placements of dropout
layer(s). For instance, the top left and bottom left panels of Figure 3 show
the prediction intervals associated with eight held-out test samples obtained
when two dropout layers were included - one inside the LSTM and one just
before the final prediction layer. In contrast, the top right and bottom right
panels of Figure 3 show the prediction intervals associated with the same test
samples obtained when the second dropout layer was removed from the foregoing
LSTM architecture. Observe how the number and placement of dropout layers
influence the variability in the width of these intervals. If we wish to
control the width variability, the placement of the dropout layer becomes a
tuning parameter and further increases the hyperparameter search space.

Figure 1: Pectin PUL Figure 2: Frequency distribution for the various
substrates Figure 3: Boxplots showing the Predictions superimposed with the
ground truth value from Heavy and Low Just LSTMs

We empirically show that our MEnKF-ANN approximator, trained on the logit
transformed LSTM-estimated probabilities as the response and the embedding of
the sequences obtained from LSTM and Doc2Vec operations as two types of
features produce more stable prediction intervals regardless of the location
of the dropout layer in the original LSTM.

##  3 Background

This section offers a brief overview of Kalman Filters and Ensemble Kalman
Filters and discusses how these methods have been used to train NNs and
approximate DLs. For an extensive discussion on KF and EnKF technique, we
direct the audience to Katzfuss et al. (2016) and Evensen (2003),
respectively.

###  3.1 Linear State Space Model & Kalman Filter

Consider a linear Gaussian state-space model given by

$\displaystyle y_{t}=H_{t}x_{t}+\epsilon_{t},\ \
\epsilon_{t}\sim\mathcal{N}_{m_{t}}(0,R_{t})$ (1) $\displaystyle
x_{t}=M_{t}x_{t-1}+\eta_{t},\ \ \eta_{t}\sim\mathcal{N}_{n}(0,Q_{t})$ (2)

where $y_{t}$ is the $m_{t}$ dimensional observation vector at time step $t$,
$x_{t}$ is the $n$ dimensional state variable at that time, $H_{t}$ and
$M_{t}$ denote the observation and the state transition matrices.

Assume that the filtering distribution of the state vector at $t-1$ is given
by

$x_{t-1}|y_{1{:}t-1}\sim\mathcal{N}(\hat{\mu}_{t-1},\hat{\Sigma}_{t-1}),$ (3)

KF computes the forecast distribution at $t$ using (2) as

$\displaystyle
x_{t}|y_{1{:}t-1}\sim\mathcal{N}(\tilde{\mu}_{t},\tilde{\Sigma}_{t})$
$\displaystyle\tilde{\mu}_{t}:=M_{t}\hat{\mu}_{t-1},$
$\displaystyle\tilde{\Sigma}_{t}:=M_{t}\hat{\Sigma}_{t-1}M_{t}^{{}^{\prime}}+Q_{t}$
(4)

Once the measurement at time step $t$ becomes available, the joint
distribution of $(x_{t},y_{t})$ is given by

$\begin{pmatrix}x_{t}\\\ y_{t}\end{pmatrix}\bigg{|}y_{1{:}t-1}\sim
N\left(\begin{pmatrix}\tilde{\mu}_{t}\\\
H_{t}\tilde{\mu}_{t}\end{pmatrix},\begin{pmatrix}\tilde{\Sigma}_{t}&\tilde{\Sigma}_{t}H_{t}^{{}^{\prime}}\\\
H_{t}\tilde{\Sigma}_{t}&H_{t}\tilde{\Sigma}_{t}H_{t}^{{}^{\prime}}+R_{t}\end{pmatrix}\right)$
(5)

Then the updated filtering distribution is
$x_{t}|y_{1{:}t}\sim\mathcal{N}(\hat{\mu}_{t},\hat{\Sigma}_{t})$ where
$\hat{\mu}_{t}$ and $\hat{\Sigma}_{t}$ are given by

$\displaystyle\hat{\mu_{t}}:=\tilde{\mu_{t}}+K_{t}(y_{t}-H_{t}\tilde{\mu_{t}}),$
$\displaystyle\hat{\Sigma_{t}}:=(I_{n}-K_{t}H_{t})\tilde{\Sigma}_{t},$ (6)

with
$K_{t}:=\tilde{\Sigma}_{t}H_{t}^{{}^{\prime}}(H_{t}\tilde{\Sigma_{t}}H_{t}^{{}^{\prime}}+R_{t})^{-1}$
being the Kalman Gain Matrix. For large $n$ and $m_{t}$, computing the
matrices in (3.1) is computationally expensive and often leads to numeric
instability.

###  3.2 Ensemble Kalman Filter

The idea of EnKF is to take an ensemble of size $N$ from the filtering
distribution at $t-1$. This ensemble is denoted as $\hat{x}^{(0)}_{t-1},\ \
\hat{x}^{(1)}_{t-1},...,\hat{x}^{(N)}_{t-1}\sim\mathcal{N}_{n}\left(\hat{\mu}_{t-1},\hat{\Sigma}_{t-1}\right)$.
In the forecast step of EnKF, (2) is applied to the ensemble members to obtain
their evolution from $t-1$ to $t$. That is

$\tilde{x}^{(i)}_{t}=M_{t}\hat{x}_{t-1}^{(i)}+\eta_{t}^{(i)},\ \
\eta_{t}^{(i)}\sim\mathcal{N}(0,Q_{t}),\ \ \ \ i=1,\ldots,N$ (7)

It can also be shown that
$\tilde{x}^{(i)}_{t}\sim\mathcal{N}(\tilde{\mu}_{t},\tilde{\Sigma}_{t})$.
Similar to the update step of the Kalman Filter, all the members of this
ensemble must be updated when the measurement at time step t becomes
available. To update these ensemble members, first, a sample is obtained for
the measurement error sequences that is
$\epsilon_{t}^{(1)},\epsilon_{t}^{(2)},...,\epsilon_{t}^{(N)}\sim\mathcal{N}_{m_{t}}(0,R_{t})$.
Then using these simulated measurement errors $N$ perturbed observations
$\tilde{y_{t}}^{(1)},\tilde{y_{t}}^{(2)},\ldots,\tilde{y_{t}}^{(N)}$ are
obtained using
$\tilde{y_{t}}^{(i)}=H_{t}\tilde{x}^{(i)}_{t}+\epsilon_{t}^{(i)}$. Since the
joint distribution of ($\tilde{x}^{(i)}_{t}$, $\tilde{y_{t}}^{(i)}$) is the
same as in (5), the updating equations are obtained by shifting the forecasted
ensemble in (7) as follows

$\hat{x}^{(i)}_{t}=\tilde{x}_{t}^{(i)}+K_{t}(y_{t}-\tilde{y}^{(i)}_{t}),\ \ \
\ i=1,\ldots,N$ (8)

It can be easily shown that
$\hat{x}^{(i)}_{t}\sim\mathcal{N}_{n}(\hat{\mu}_{t},\hat{\Sigma}_{t})$. The
computational benefit comes from the fact that instead of computing the Kalman
gain matrix in (8) explicitly, the sample covariance matrix of the forecasted
ensemble ($\tilde{S_{t}}$, say) is used to estimate the Kalman Gain matrix as
$\hat{K_{t}}:=\tilde{S}_{t}H_{t}^{{}^{\prime}}(H_{t}\tilde{S_{t}}H_{t}^{{}^{\prime}}+R_{t})^{-1}$.

###  3.3 KF and EnKF for Deep Learners

Although conventional KF is only suitable for estimating parameters in linear
state-space models, several extensions have been proposed to generalize KF in
nonlinear settings. For instance, Rivals and Personnaz (1998) used extended KF
to train feed-forward NN. Wan and Van Der Merwe (2000) introduced the
unscented KF that better approximates nonlinear systems while making it
amenable to the KF framework. Anderson (2001) used the concept of state
augmentation that offered a generic method to handle nonlinearity in state-
space models via the KF framework. Iglesias et al. (2013) utilized this state
augmentation technique to develop a generic method to train ANNs. They derived
the state-augmented KF’s forecast and updated equations in ANNs, thereby
providing the algebraic framework to train DLs using the Ensemble Kalman
Filters approach. These equations were subsequently used by Yegenoglu et al.
(2020) to train a Convolutional Neural Network using EnKF. Furthermore, Chen
et al. (2019) also used the updating equations in Iglesias et al. (2013) to
train a single hidden layer ANN and demonstrated how using state augmentation
one can estimate the measurement error variance. State-augmented EnKF
formulation also estimated parameters in LSTMs (Chen et al., 2018).

All the foregoing models offer techniques to estimate parameters of complex
nonlinear DLs using the EnKF framework. However, they are unsuitable when we
have multiple feature representations. We want to approximate a DL with a
multi-arm ANN trained via EnKF, as discussed in section 2.

##  4 Methodology

First, we offer a generic construction of the proposed MEnKF-ANN procedure and
describe how this method could be deployed to solve the problem in section 2.
We will use the following notations. $Y\in\mathcal{R}$ is our target response.
We have a total of $m=\sum_{t=1}^{T}m_{t}$ training instances, with $m_{t}$
being the number of training data points in the $t^{th}$ batch.
$v_{t}^{f}\in\mathcal{R}^{p}$ and $v_{t}^{g}\in\mathcal{R}^{q}$ denote two
different representations of the features (possibly of different dimensions)
for the $t^{th}$ batch of data. Consider two ANNs, denoted by $f$ and $g$.
Assume that the neural networks f and g have $n_{f}$, $n_{g}$ number of
learnable parameters. For illustrative purposes, we will assume $n_{f}=n_{g}$.
If $n_{f}\neq n_{g}$, we can use suitable padding when updating the weights.
In the $t^{th}$ batch of data, we assign the feature sets $v_{t}^{f}$ and
$v_{t}^{g}$ to networks $f$ and $g$, respectively. We denote $w_{t}^{f}$ and
$w_{t}^{g}$ to be the updated weights for the neural network $f$ and $g$,
respectively.

###  4.1 Matrix Kalman Filter based Multi-arm ANN

Consider the state matrix, $X_{t}$, associated with the $t^{th}$ batch of data
given by

$X_{t}^{(m_{t}+n_{g}+1)\times 2}=\begin{bmatrix}f(v_{t}^{f},w_{t}^{f}),\ \
g(v_{t}^{g},w_{t}^{g})\\\ w_{t}^{f},\ \ w_{t}^{g}\\\ 0,\ \ a_{t}\end{bmatrix}$
(9)

where $a_{t}$ and $b_{t}$ are real-valued scalar parameters. Define
$H_{t}^{m_{t}\times(m_{t}+n_{g}+1)}=[I_{m_{t}},0_{m_{t}\times(n_{g}+1)}]$ and
$G_{t}^{2\times 1}=[1-\sigma(a_{t}),\ \ \sigma(a_{t})]^{T}$ where
$\sigma(.):\mathcal{R}\rightarrow{[0,1]}$, with the sigmoid function being a
popular choice of $\sigma(.)$. Additionally, define
$\Theta_{t-1}=I_{m_{t}+n_{g}+1}$ and $\psi_{t-1}=I_{2}$. We are now in a
position to define the Matrix Kalman Filter. The measurement equation is given
by:

$Y_{t}=H_{t}X_{t}G_{t}+\epsilon_{t}$ (10)

with the state evolution equation being

$X_{t}=\Theta_{t-1}X_{t-1}\psi_{t-1}+\eta_{t}$ (11)

Writing in $vec$ format, (11) becomes

$x_{t}=vec(X_{t})=(\psi^{T}_{t-1}\otimes\Theta_{t-1})vec(X_{t-1})+vec(\eta_{t})$
(12)

Now letting $\phi_{t-1}=\psi^{T}_{t-1}\otimes\Theta_{t-1}$ and
$\tilde{\eta}_{t}=vec(\eta_{t})$ we get from (12)

$x_{t}=\phi_{t-1}x_{t-1}+\tilde{\eta}_{t}$ (13)

(10) can similarly be compactified as

$y_{t}=\mathcal{H}_{t}x_{t}+\epsilon_{t}$ (14)

where $\mathcal{H}_{t}=G^{T}_{t}\otimes H_{t}$. Observe that(14) and (13) have
the same form as the standard representation of linear state space model
described in (1) and (2). Therefore, we can get the matrix state space model’s
solution by converting it to the vector state space model and then using EnKF
to approximate the updating equations. We direct the audience to (Choukroun et
al., 2006) for more details on Matrix Kalman Filters.

###  4.2 Interpreting MEnKF-ANN and a Reparametrization

The above construction of $X_{t}$, $H_{t}$, and $G_{t}$ performs automatic
model averaging while training. First, consider the matrix multiplication of
$H_{t}X_{t}$ from (10). This would be a $m_{t}\times 2$ dimensional matrix in
which the first column is the prediction, for the $t^{th}$ batch, from the
neural network $f$ and the second column is the prediction from the neural
network $g$. Post multiplication by $G_{t}$ would take the weighted average of
each row in $H_{t}X_{t}$ where the weights are defined inside the $G_{t}$
matrix. Now consider the matrix multiplication of $H_{t}X_{t}G_{t}$ from (10)

$\displaystyle H_{t}X_{t}G_{t}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}f(v_{t}^{f},w_{t}^{f}),\ \
g(v_{t}^{g},w_{t}^{g})\end{bmatrix}\begin{bmatrix}1-\sigma(a_{t})\\\
\sigma(a_{t})\end{bmatrix}$ (15) $\displaystyle=$
$\displaystyle\begin{bmatrix}(1-\sigma(a_{t}))f(v_{t}^{f},w_{t}^{f})+\sigma(a_{t})g(v_{t}^{g},w_{t}^{g})\end{bmatrix}$

(15) clearly demonstrates how our construction explicitly performs model
averaging across the batches with $1-\sigma(a_{t})$ and $\sigma(a_{t})$ being
the convex weights allocated to the ANNs $f$ and $g$, respectively.

Although the foregoing construction connects Matrix KF formulation with multi-
arm ANN and performs explicit model averaging, it suffers from a computational
bottleneck. Using (13) and (14) the estimated Kalman Gain Matrix would be
$K_{t}=\tilde{S}_{t}\mathcal{H}_{t}^{T}(\mathcal{H}_{t}\tilde{S}_{t}\mathcal{H}_{t}^{T}+\sigma^{2}_{y}I_{m_{t}})^{-1}$
. However, in the above parameterization we have $G_{t}=[1-\sigma(a_{t}),\ \
\sigma(a_{t})]^{T}$ and $\mathcal{H}_{t}=G^{T}_{t}\otimes H_{t}$. This would
require computation of the estimated Kalman Gain matrix for each member in
EnKF since, at any given iteration of our MEnKF-ANN, we have an $a_{t}$ for
each member of the ensemble. Thus computation complexity associated with the
Kalman Gain computation increases linearly with the size of the ensemble in
the above parametrization of the MEnKF-ANN.

To alleviate this computational bottleneck, consider the following
parametrization:

$\displaystyle X_{t}=\begin{bmatrix}(1-\sigma(a_{t}))f(v_{t}^{f},w_{t}^{f}),\
\sigma(a_{t})g(v_{t}^{g},w_{t}^{g})\\\ w_{t}^{f},\ \ w_{t}^{g}\\\ 0,\ \
a_{t}\end{bmatrix}$ (16)

and $G_{t}=[1,\ \ 1]^{T}$. We still have explicit model averaging in the
measurement equation, i.e.,

$\displaystyle
H_{t}X_{t}G_{t}=\begin{bmatrix}(1-\sigma(a_{t}))f(v_{t}^{f},w_{t}^{f})+\sigma(a_{t})g(v_{t}^{g},w_{t}^{g})\end{bmatrix}$
(17)

but $\mathcal{H}_{t}$ does not depend on $a_{t}$. Therefore the matrix
products for the Kalman Gain computation can now be computed once for each
batch.

Turning to the variance parameter in the measurement equation (14). Assume
$\epsilon_{t}\sim\mathcal{N}_{m_{t}}(0,\nu^{2}_{y}I_{m_{t}})$ To estimate
$\nu^{2}_{y}$, we augment the state vector as follows:

$\displaystyle X_{t}^{(m_{t}+n_{g}+2)\times
2}=\begin{bmatrix}(1-\sigma(a_{t}))f(v_{t}^{f},w_{t}^{f}),\
\sigma(a_{t})g(v_{t}^{g},w_{t}^{g})\\\ w_{t}^{f},\ \ w_{t}^{g}\\\ 0,\ \
a_{t}\\\ 0,\ \ b_{t}\end{bmatrix}$ (18)

where $\nu^{2}_{y}=\log(1+e^{b_{t}})$ and $H_{t}$ in (10) now becomes
$[I_{m_{t}},0_{m_{t}\times(n_{g}+2)}]$. We used a softplus transformation of
$\nu^{2}_{y}$, instead of the usual log transformation for computational
stability.

###  4.3 Connecting MEnKF-ANN with DL

Recall that our dataset consists of string-valued gene sequences associated
with experimentally determined PULs, with the response being the carbohydrate
substrates utilized by the said microbe. Since we consider only two categories
of PULs, we have a binary classification problem. An LSTM trained with a
binary cross-entropy loss is the approximand DL in our case. Suppose $p$ is
the probability of observing a sample of a particular category. In that case,
the trained LSTM produces $\hat{p}$ for each training instance, along with an
embedding of the associated gene sequences. Our MEnKF-ANN approximator uses
$logit(\hat{p})$ as the target response. The LSTM embedding of the gene
sequences is fed into one arm of the approximator while the other arm ingests
Doc2Vec encoding of the gene sequences. Thus, our MEnKF-ANN approximates the
probabilities estimated by an LSTM. The convex weights $\sigma(a)$ ascertain
which embedding has more predictive power. Clearly, MEnKF-ANN operates as a
model stacker, and the predictive uncertainty interval that it produces, by
default, around its target approximand quantifies how well simpler ANNs,
fitted without backpropagation, can approximate a deep learner.

To initialize the ensemble in the approximator, we draw the members in the
state vector (18) from
$\mathcal{N}_{2(m_{t}+n_{g}+2)}(\mathbf{0},\nu^{2}_{x}I)$, where $\nu^{2}_{x}$
is a tuning parameter that plays a key role in controlling the spread of the
ensemble members and the dimension of $I$ matches with the dimension of normal
distribution. Following Chen et al. (2018, 2019), we assume the state
transition is deterministic, i.e., $x_{t}=\phi_{t-1}x_{t-1}$ and hence we do
not have the variance parameter corresponding to $\tilde{\eta}$ in the
augmented state vector. When we reach the $t^{th}$ batch of data, for the
$i^{th}$ member in the ensemble ($i=1,2,...,N$), we update each element in the
augmented state vector $w^{f,(i)}_{t}$, $w^{g,(i)}_{t}$, $a^{(i)}_{t}$,
$b^{(i)}_{t}$ using the updating equation (8) suitably modified to handle
deterministic state transition.

##  5 Simulations

We conducted extensive simulations to assess how well our MEnKF-ANN can
approximate an LSTM binary classifier. This simulation exercise aims to
demonstrate that our MEnKF-ANN is not only “adequate” in approximating the
probabilities produced by LSTM but can also capture the “true” probabilities
that generate binary labels. We compute the coverage and width of the
prediction intervals of the target probabilities in the test set to assess the
“adequacy” of the approximator. Then, we compare this coverage and width with
those computed directly via an LSTM trained with MC dropout. Admittedly, the
prediction intervals obtained from the latter are different from those
computed from MEnKF-ANN. However, if the ground truth probabilities are known,
an adequate approximator should be able to achieve near-nominal coverage when
the approximand is not misspecified.

Our simulation strategy mimics the focal dataset and uses the gene sequences
associated with the original PULs to generate labels. As mentioned above, we
extracted $\hat{p}$ from the LSTM trained on the original dbCAN-PUL data. We
call this LSTM the true LSTM. We consider $\hat{p}$ the true probabilities for
synthetic data generation. We then use noisy copies of $\hat{p}$ to generate a
synthetic label in the following way: generate
$logit(\tilde{p}_{i}^{(j)})=logit(\hat{p}_{i})+\epsilon_{i}^{*(j)},\;i=1,2,...,m,j=1,2,...,J$,
where $J$ is the number of the simulated dataset and $m$, is the number of
data points in each simulated set, the perturbation $\epsilon_{i}^{*(j)}$ are
iid Normal(0,$0.01^{2}$). We generate synthetic labels $\tilde{Y}$ by
thresholding $\tilde{p}_{i}^{(j)}$ at 0.5, i.e
$\tilde{Y}_{i}^{(j)}=I(\tilde{p}_{i}^{(j)}>0.5)$. Then the simulated dataset
consists of $D^{(j)}=\\{\bm{F},\tilde{Y}^{(j)},\;j=1,2,...,J\\}$, where
$\bm{F}$ is the set of original gene sequences from dbCAN-PUL.

Now in each $D^{(j)}$, we train a second LSTM (with two dropout layers) and
extract $\tilde{\tilde{p}}_{i}^{(j)},i=1,2,...,m$ and the embedding of the
gene sequences. We call these LSTMs, trained on $D^{(j)}$, the fitted LSTMs.
Note that the embeddings from fitted LSTMs could potentially be different from
those obtained from the true LSTM. We denote the embedding from fitted LSTMs
by $v_{i}^{(j),f},\;j=1,2,...J$. Our MEnKF-ANN is constructed to approximate
the fitted LSTMs. To that end, the approximator uses
$logit(\tilde{\tilde{p}}_{i}^{(j)})$ as the target response. $v_{i}^{(j),f}$
are supplied as features to one arm of the ANN, the other arm ingests
$v_{i}^{(j),g}$ \- the Doc2Vec embedding of $\bm{F}$. Once the MEnKF-ANN is
trained, we use a hold-out set in each simulated dataset to generate
predictive probabilities from the forecast distribution for each member in the
KF ensemble and compute the empirical 95% predictive interval at $logit^{-1}$
scale. To measure the adequacy of MEnKF-ANN, we compute the proportion of
times the foregoing predictive interval contains $\hat{p}$ in held-out test
data. We expect this coverage to be close to the nominal 95%, and the average
width of these intervals should not be greater than 0.5. Additionally, observe
that the data-generating model uses LSTM embedding of $F$; hence, using
Doc2Vec embedding as input is misspecification. Consequently, we expect the
average model weight associated with $v^{f}$ to be larger than $v^{g}$. Table
1 shows the performance of MEnKF-ANN in terms of coverage, the average width
of prediction intervals, and average LSTM weight under two specifications of
ensemble size ($N$) and initial ensemble variance $(\nu^{2}_{x})$. To compare
these results, we offer the coverage and average width of the prediction
intervals when both the dropout layers are activated in the fitted LSTM during
the prediction phase in Table 2. Observe how MEnKF-ANN recovered the true
probabilities even better than the correctly specified LSTM with dropout. The
average interval widths obtained from MEnKF-ANN are also lower than those from
the fitted LSTM. These demonstrate the adequacy of MEnKF-ANN in approximating
the target DL. Additionally, we observe that the average LSTM model weight is
$\approx 1$ indicating the ability of our approximator to identify the
correctly specified data-generating model. Figure 4 shows the histogram of the
predictive samples obtained from the ensemble members for eight test samples
in a randomly chosen replicate. The red vertical line denotes the true logits,
and the green vertical lines show the fences of the 95% prediction interval.

Now, to demonstrate a situation where MEnKF-ANN is “inadequate”, we supply the
approximator with a completely different feature set representation. Instead
of using the LSTM embedding $v^{f}$, we use word2vec embedding of each gene in
the predictor string and take the arithmetic average of these word2vec
embeddings to represent the entire sequence. We denote this feature set by
$\tilde{v}^{f}$ and then train the MEnKF-ANN using $\tilde{v}^{f}$ and $v^{g}$
as the features and $logit(\tilde{\tilde{p}}^{(j)})$ as the target response.
Evidently, MEnKF-ANN is highly misspecified. Table 3 reports the coverage and
average width of the prediction interval obtained from this model. Observing
the huge width of the intervals essentially invalidates the point prediction.
Such large width indicates that MEnKF-ANN may not approximate the target DL.
Therefore, we caution against using the coverage and width metrics to assess
the “adequacy” of the fitted LSTM itself.

Table 1: Performance of MEnKF-ANN using LSTM embedding and Doc2Vec  $N$ | $\nu^{2}_{x}$ | Coverage | Width | LSTM weight
---|---|---|---|---
216 | 16 | 90.25$\%$ | 0.33 | 0.9997
216 | 32 | 89.25$\%$ | 0.32 | 0.9999
Table 2: Coverage and width of prediction intervals obtained from fitted LSTM with two dropout layers  Rate | Reps | Coverage | Width
---|---|---|---
0.5 | 50 | 81.25$\%$ | 0.53
0.5 | 200 | 84.50$\%$ | 0.56
Table 3: Performance of MEnKF-ANN using Word2Vec and Doc2Vec  $N$ | $\nu^{2}_{x}$ | Coverage | Width | Word2Vec weight
---|---|---|---|---
216 | 16 | 96.25$\%$ | 0.83 | 0.9155
216 | 32 | 94.25$\%$ | 0.84 | 0.9787
Figure 4: True Logits Superimposed on Predicted Logits from MEnKF-ANN using
LSTM and Doc2Vec embedding

##  6 Application

Recall that our focal dataset consists of $n=74$ samples belonging to Xylan
and Pectin. However, training an LSTM on a small sample size would require
aggressive regularization, even with this reduced label space. Therefore, we
draw on an extensive collection of unlabelled data containing gene sequences
associated with CAZyme gene clusters (CGC) computationally predicted from
genomic data (Huang et al., 2018; Zheng et al., 2023). Although this
unlabelled data contain approximately 250K CGC gene sequences, unlike
experimentally characterized PULs, these sequences do not have known
carbohydrate substrate information. They hence cannot be directly used for
classification purposes. We, therefore, use this unlabelled dataset to learn
the word2vec embeddings of each gene appearing in the unlabelled dataset.
These embeddings are then used to initialize the embedding layer of the target
LSTM classifier.

Turning to the labeled dataset, instead of performing full cross-validation,
we resort to subsampling procedure (Politis et al., 1999). We take a subsample
of sixty-six instances for training and hold eight instances for testing
purposes. The subsample size ($b$) is chosen such that $b(n)/n\approx
8\sqrt{n}/n\rightarrow 0,\;$ as $n\rightarrow\infty$. Although the subsampling
theory requires generating $\begin{pmatrix}n\\\ b\end{pmatrix}$ replicates,
the computational cost for generating $\approx 10^{11}$ replicates, in our
case, is prohibitive. Instead, we generate 50 independently subsampled
replicates comprising training and testing sets of sizes 66 and 8,
respectively. In each replication, an LSTM with the same architecture is
trained on the foregoing 66 training instances. Under this scheme, the
probability that the $i^{th}$ instance in our dataset appears at least once in
the test set is $\approx 99.6\%$.

The LSTM-estimated probabilities of observing a Pectin substrate are extracted
from each replicate. These probabilities are logit transformed and used as the
target response for our MEnKF-ANN approximator. We feed the LSTM embedding and
Doc2Vec embedding of the gene sequences into the two arms of the approximator
along with the foregoing logit-transformed estimated probabilities. We then
generate predictions on the held-out test data points in each replicate.
Finally, we compare the LSTM-prediction of probabilities with those generated
by MEnKF-ANN generated predictions. The average MAE and the proportion of
times a 95% prediction interval contains the LSTM-generated predictions in the
held-out data set, under two different MEnKF-ANN hyperparameter choices are
shown in Table 4 indicating that our approximator can be adequately used to
generate the predictions. We do not report the LSTM weights estimated by
MEnKF-ANN because, as we observed in the simulation (Table 1), the
approximator overwhelmingly prefers the LSTM embeddings. Figure 5 shows the
scatter plot of MEnKF-ANN-predicted and LSTM-predicted probabilities for the
held-out data across 50 replicates. Figure 6 shows the boxplots associated
with MEnKF-ANN predictions for the same set of test samples for which LSTM
generated prediction boxplots were shown in the left column of Figure 3.
Evidently, MEnKF-ANN can adequately approximate the target LSTM.

Turning to the stability of prediction intervals, Table 5 shows the average
width of the 95% prediction intervals obtained under two configurations of
LSTM and their respective MEnKF-ANN approximators. LSTM1 has two dropout
layers (one in the LSTM layer and one before the final prediction layer) with
a 50% dropout rate and 200 replicates. LSTM2 has one dropout layer (in the
LSTM layer) with a 50% dropout rate and 200 replicates. MEnKF-ANN11
approximates LSTM1 with 216 ensemble members and $\nu^{2}_{x}=16$, MEnKF-ANN12
also approximates LSTM1, but now with 216 ensemble members and
$\nu^{2}_{x}=32$. Similarly, MEnKF-ANN21 and MEnKF-ANN22 approximates LSTM2
with 216 ensemble members and $\nu^{2}_{x}=16$ and $\nu^{2}_{x}=32$,
respectively. Observe that the variation in the average width between LSTM1
and LSTM2 is considerably higher than the variation between MEnKF-ANN11 and
MEnKF-ANN21 or between MEnKF-ANN12 and MEnKF-ANN22. This indicates that the
approximator produces more stable prediction intervals than obtaining
prediction by activating the dropout layer during prediction.

Finally, we demonstrate how MEnKF-ANN can naturally handle two predictive
models with potentially different feature sets. This situation is relevant
because, owing to the small sample size, we can train a shallow learner (ANN
with backpropagation, for instance) that takes Doc2Vec representation of gene
sequences as predictors to estimate the probabilities of observing the Pectin
substrate. Now, we can average the probabilities estimated by the LSTM
($\hat{p}_{LSTM}$, say) and ANN ($\hat{p}_{ANN}$, say) to produce a model-
averaged estimated probability of observing Pectin ($\hat{\bar{p}}$, say).
However, how would we attach uncertainty to $\hat{\bar{p}}$ ? The multi-arm
construction of MEnKF-ANN provides a natural solution in this situation. We
supply, as described in the foregoing sections, LSTM embeddings and Doc2Vec
embeddings to the two arms of MEnKF-ANN but use $logit(\hat{\bar{p}})$ as the
target response here. Thus MEnKF-ANN is now approximating the average output
of two primary models. These primary models are trained on the same response
variable but use two different representations of features. Table 6 shows the
performance of MEnKF-ANN in this situation for some combinations of $N$ and
$\nu^{2}_{x}$. The coverage is measured with respect to $\hat{\bar{p}}$ on the
test sets. Although the average width and MAE are larger than those reported
in Table 4, we observe that the LSTM weights $\approx$ 0.5, which is what we
would expect because MEnKF-ANN is seeing equally weighted outputs from LSTM
and ANN.

Table 4: Performance of MEnKF-ANN using LSTM embedding and Doc2Vec for dbCAN-PUL data  $N$ | $\nu_{x}^{2}$ | Coverage | Width | MAE | CPU Time
---|---|---|---|---|---
216 | 16 | 90.50$\%$ | 0.1024 | 0.0200 | 2.39 mins
216 | 32 | 85.50$\%$ | 0.0850 | 0.0161 | 3.67 mins
Table 5: Comparison of the average width of prediction interval LSTM + MC Dropout and MEnKF-ANN approximator for each LSTM  Target model | Average Width | Approximator | Average Width
---|---|---|---
LSTM1 | 0.492 | MEnKF-ANN11 | 0.102
|  | MEnKF-ANN12 | 0.085
LSTM2 | 0.371 | MEnKF-ANN21 | 0.119
|  | MEnKF-ANN22 | 0.108
Table 6: Performance of MEnKF-ANN trained on the averaged probability of LSTM and shallow ANN  $N$ | $\nu_{x}^{2}$ | Coverage | Width | MAE | LSTM weight
---|---|---|---|---|---
433 | 2.0 | 85.50$\%$ | 0.2249 | 0.0615 | 0.4954
433 | 4.0 | 78.00$\%$ | 0.1978 | 0.0684 | 0.5006
Figure 5: Scatterplot of First LSTM Predicted Probabilities vs. EnKF Predicted
Probabilities Figure 6: Boxplots showing the MEnKF-ANN Predictions
superimposed with the ground truth value for Heavy and Low Dropout

##  7  Discussion

State-augmented Kalman Filter and its variants provide a gradient-free method
that can be extended to approximate popular neural network-based deep learners
for regression and classification tasks. In this article, we have developed a
Matrix Ensemble Kalman Filter-based multi-arm neural network to approximate an
LSTM. We have demonstrated that this technique adequately approximates the
target DL regarding coverage and the average width of the prediction interval.
We have also demonstrated how the in-built model averaging capability can be
leveraged to attach uncertainty to the averaged predictions generated by two
different models. Our simulations suggest that by using an explicit model
averaging construction, our approximator can also identify its target
approximand. We have also observed that the prediction intervals generated by
the approximator are less sensitive to the location of dropout layers and
hence provide more stable prediction intervals than obtaining predictions by
activating the dropout layers within the DL itself. Admittedly, our procedure
requires an additional round of training, but its fast computation time (see
Table 4), along with its ability to emulate the approximand, adequately
compensate for that. We have also deployed our approximator on a highly
accessed database, dbCAN-PUL, to attach uncertainty to the predicted
probabilities produced by (a) the primary LSTM model and (b) an ensemble of
LSTM and ANN models. The primary LSTM and ANN models were trained to classify
two carbohydrate substrates using the gene sequences characterized by the PULs
of the gut microbiome. We anticipate this technique will be helpful to domain
experts in assessing the reliability of predictions generated by deep learners
or an ensemble of learners.

In the future, we propose to expand our model to handle more than two classes.
This would enable us to utilize the information in the dbCAN-PUL database
better. Another possible direction is to develop an analog of MEnKF-ANN that
can directly handle binary data. Although the KF technique crucially requires
Gaussianity assumption, but Fasano et al. (2021) recently developed an
extension of the KF method that can handle binary responses. We are actively
investigating how this technique can be adapted to our MEnKF-ANN framework.

##  8 Competing interests

No competing interest is declared.

##  9 Author contributions statement

V.P., Y.Z., and S.G. conceived the models and experiment(s), and V.P.
conducted the experiment(s). V.P. and S.G. analyzed the results. Y.Yan and
Y.Yin contributed to the training data. V.P. drafted the manuscript. All
authors reviewed the manuscript. Y. Yin secured the funding.

##  10 Acknowledgments

The authors thank the anonymous reviewers for their valuable suggestions. This
work is supported in part by funds from the National Institute of Health (NIH:
R01GM140370, R21AI171952) and the National Science Foundation (NSF:
CCF2007418, DBI-1933521). In addition, we thank the lab members for their
helpful discussions. This work was partially completed utilizing the Holland
Computing Center of the University of Nebraska–Lincoln, which receives support
from the Nebraska Research Initiative.

## References

  * Anderson [2001] J. L. Anderson.  An ensemble adjustment kalman filter for data assimilation.  _Monthly weather review_ , 129(12):2884–2903, 2001.
  * Ausland et al. [2021] C. Ausland, J. Zheng, H. Yi, B. Yang, T. Li, X. Feng, B. Zheng, and Y. Yin.  dbcan-pul: a database of experimentally characterized cazyme gene clusters and their substrates.  _Nucleic Acids Research_ , 49(D1):D523–D528, 2021\.
  * Badjatiya et al. [2017] P. Badjatiya, S. Gupta, M. Gupta, and V. Varma.  Deep learning for hate speech detection in tweets.  In _Proceedings of the 26th international conference on World Wide Web companion_ , pages 759–760, 2017.
  * Bjursell et al. [2006] M. K. Bjursell, E. C. Martens, and J. I. Gordon.  Functional genomic and metabolic studies of the adaptations of a prominent adult human gut symbiont, bacteroides thetaiotaomicron, to the suckling period.  _Journal of biological chemistry_ , 281(47):36269–36279, 2006.
  * Blundell et al. [2015] C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra.  Weight uncertainty in neural network.  In _International conference on machine learning_ , pages 1613–1622. PMLR, 2015.
  * Chen et al. [2018] C. Chen, X. Lin, and G. Terejanu.  An approximate bayesian long short-term memory algorithm for outlier detection.  In _2018 24th International Conference on Pattern Recognition (ICPR)_ , pages 201–206. IEEE, 2018.
  * Chen et al. [2019] C. Chen, X. Lin, Y. Huang, and G. Terejanu.  Approximate bayesian neural network trained with ensemble kalman filter.  In _2019 International Joint Conference on Neural Networks (IJCNN)_ , pages 1–8. IEEE, 2019.
  * Choukroun et al. [2006] D. Choukroun, H. Weiss, I. Y. Bar-Itzhack, and Y. Oshman.  Kalman filtering for matrix estimation.  _IEEE Transactions on Aerospace and Electronic Systems_ , 42(1):147–159, 2006.
  * Evensen [2003] G. Evensen.  The ensemble kalman filter: Theoretical formulation and practical implementation.  _Ocean dynamics_ , 53(4):343–367, 2003.
  * Fasano et al. [2021] A. Fasano, G. Rebaudo, D. Durante, and S. Petrone.  A closed-form filter for binary time series.  _Statistics and Computing_ , 31(4):47, 2021.
  * Flint et al. [2012] H. J. Flint, K. P. Scott, S. H. Duncan, P. Louis, and E. Forano.  Microbial degradation of complex carbohydrates in the gut.  _Gut microbes_ , 3(4):289–306, 2012.
  * Fortuin et al. [2021] V. Fortuin, A. Garriga-Alonso, S. W. Ober, F. Wenzel, G. Rätsch, R. E. Turner, M. van der Wilk, and L. Aitchison.  Bayesian neural network priors revisited.  _arXiv preprint arXiv:2102.06571_ , 2021.
  * Gal and Ghahramani [2016] Y. Gal and Z. Ghahramani.  Dropout as a bayesian approximation: Representing model uncertainty in deep learning.  In _international conference on machine learning_ , pages 1050–1059. PMLR, 2016.
  * Hinton and Van Camp [1993] G. E. Hinton and D. Van Camp.  Keeping the neural networks simple by minimizing the description length of the weights.  In _Proceedings of the sixth annual conference on Computational learning theory_ , pages 5–13, 1993.
  * Huang et al. [2018] L. Huang, H. Zhang, P. Wu, S. Entwistle, X. Li, T. Yohe, H. Yi, Z. Yang, and Y. Yin.  dbcan-seq: a database of carbohydrate-active enzyme (cazyme) sequence and annotation.  _Nucleic Acids Research_ , 46(D1):D516–D521, 2018\.
  * Iglesias et al. [2013] M. A. Iglesias, K. J. Law, and A. M. Stuart.  Ensemble kalman methods for inverse problems.  _Inverse Problems_ , 29(4):045001, 2013.
  * Julier and Uhlmann [2004] S. J. Julier and J. K. Uhlmann.  Unscented filtering and nonlinear estimation.  _Proceedings of the IEEE_ , 92(3):401–422, 2004\.
  * Kaoutari et al. [2013] A. E. Kaoutari, F. Armougom, J. I. Gordon, D. Raoult, and B. Henrissat.  The abundance and variety of carbohydrate-active enzymes in the human gut microbiota.  _Nature Reviews Microbiology_ , 11(7):497–504, 2013.
  * Katzfuss et al. [2016] M. Katzfuss, J. R. Stroud, and C. K. Wikle.  Understanding the ensemble kalman filter.  _The American Statistician_ , 70(4):350–357, 2016\.
  * Kononenko [1989] I. Kononenko.  Bayesian neural networks.  _Biological Cybernetics_ , 61(5):361–370, 1989\.
  * Koropatkin et al. [2012] N. M. Koropatkin, E. A. Cameron, and E. C. Martens.  How glycan metabolism shapes the human gut microbiota.  _Nature Reviews Microbiology_ , 10(5):323–335, 2012.
  * Liu et al. [2020] Q. Liu, Z. Hu, R. Jiang, and M. Zhou.  Deepcdr: a hybrid graph convolutional network for predicting cancer drug response.  _Bioinformatics_ , 36(Supplement_2):i911–i918, 2020.
  * Lombard et al. [2014] V. Lombard, H. Golaconda Ramulu, E. Drula, P. M. Coutinho, and B. Henrissat.  The carbohydrate-active enzymes database (cazy) in 2013.  _Nucleic acids research_ , 42(D1):D490–D495, 2014\.
  * Min et al. [2017] S. Min, B. Lee, and S. Yoon.  Deep learning in bioinformatics.  _Briefings in bioinformatics_ , 18(5):851–869, 2017.
  * Neal [2012] R. M. Neal.  _Bayesian learning for neural networks_ , volume 118.  Springer Science & Business Media, 2012.
  * Politis et al. [1999] D. N. Politis, J. P. Romano, and M. Wolf.  _Subsampling_.  Springer Science & Business Media, 1999.
  * Pudlo et al. [2022] N. A. Pudlo, K. Urs, R. Crawford, A. Pirani, T. Atherly, R. Jimenez, N. Terrapon, B. Henrissat, D. Peterson, C. Ziemer, et al.  Phenotypic and genomic diversification in complex carbohydrate-degrading human gut bacteria.  _Msystems_ , 7(1):e00947–21, 2022.
  * Rivals and Personnaz [1998] I. Rivals and L. Personnaz.  A recursive algorithm based on the extended kalman filter for the training of feedforward neural models.  _Neurocomputing_ , 20(1-3):279–294, 1998.
  * Sharifi-Noghabi et al. [2019] H. Sharifi-Noghabi, O. Zolotareva, C. C. Collins, and M. Ester.  Moli: multi-omics late integration with deep neural networks for drug response prediction.  _Bioinformatics_ , 35(14):i501–i509, 2019.
  * Srivastava et al. [2014] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov.  Dropout: a simple way to prevent neural networks from overfitting.  _The journal of machine learning research_ , 15(1):1929–1958, 2014.
  * Stewart et al. [2018] R. D. Stewart, M. D. Auffret, R. Roehe, and M. Watson.  Open prediction of polysaccharide utilisation loci (pul) in 5414 public bacteroidetes genomes using pulpy.  _BioRxiv_ , page 421024, 2018.
  * Valdes et al. [2018] A. M. Valdes, J. Walter, E. Segal, and T. D. Spector.  Role of the gut microbiota in nutrition and health.  _Bmj_ , 361, 2018.
  * Wan and Van Der Merwe [2000] E. A. Wan and R. Van Der Merwe.  The unscented kalman filter for nonlinear estimation.  In _Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373)_ , pages 153–158. Ieee, 2000.
  * Yegenoglu et al. [2020] A. Yegenoglu, K. Krajsek, S. D. Pier, and M. Herty.  Ensemble kalman filter optimizing deep neural networks: An alternative approach to non-performing gradient descent.  In _International Conference on Machine Learning, Optimization, and Data Science_ , pages 78–92. Springer, 2020.
  * Zhang et al. [2018] H. Zhang, T. Yohe, L. Huang, S. Entwistle, P. Wu, Z. Yang, P. K. Busk, Y. Xu, and Y. Yin.  dbcan2: a meta server for automated carbohydrate-active enzyme annotation.  _Nucleic acids research_ , 46(W1):W95–W101, 2018\.
  * Zheng et al. [2023] J. Zheng, B. Hu, X. Zhang, Q. Ge, Y. Yan, J. Akresi, V. Piyush, L. Huang, and Y. Yin.  dbcan-seq update: Cazyme gene clusters and substrates in microbiomes.  _Nucleic Acids Research_ , 51(D1):D557–D563, 2023\.


00footnotetext: $\bf{2020\ Mathematics\ Subject\ Classification\>}.$ 53D50,
53C25, 53C80.
Key words and phrases: $(k,\mu)^{\prime}$\- almost Kenmotsu manifolds,
Kenmotsu manifolds, $(m,\rho)$-quasi Einstein solitons.
††thanks: ∗ Corresponding author

# A note on gradient Solitons on two classes of almost Kenmotsu Manifolds

Krishnendu De∗ and Uday Chand De  Department of Mathematics, Kabi Sukanta
Mahavidyalaya, The University of Burdwan. Bhadreswar, P.O.-Angus, Hooghly, Pin
712221, West Bengal, India. ORCID iD: https://orcid.org/0000-0001-6520-4520
<EMAIL_ADDRESS>Department of Mathematics, University of Calcutta,
West Bengal, India. ORCID iD: https://orcid.org/0000-0002-8990-4609  uc-
<EMAIL_ADDRESS>
###### Abstract

The purpose of the article is to characterize gradient $(m,\rho)$-quasi
Einstein solitons within the framework of two classes of almost Kenmotsu
Manifolds. Finally, we consider an example to justify a result of our paper.

##  1 Introduction

The techniques of contact geometry carried out a significant role in
contemporary mathematics and consequently it have become well-known among the
eminent researchers. Contact geometry has manifested from the mathematical
formalism of classical mechanics. This subject matter has more than one
attachments with the alternative regions of differential geometry and
outstanding applications in applied fields such as phase space of dynamical
systems, mechanics, optic and thermodynamics. In the existing paper, we have a
look at the gradient solitons with nullity distribution which play a
functional role in coeval mathematics.

In the study of Riemannian manifolds $(M,g)$, Gray [7] and Tanno [11]
introduced the concept of _$k$ -nullity distribution $(k\in\mathbb{R})$_ and
is defined for any $p\in M$ and $k\in\mathbb{R}$ as follows:

$N_{p}(k)=\\{W\in T_{p}M:R(U,V)W=k[g(V,W)U-g(U,W)V]\\},$ (1.1)

for any $U,V\in T_{p}M$, where $R$ indicates the Riemannian curvature tensor
of type $(1,3)$.

Recently, the generalized idea of the $k$-nullity distribution, named as
_$(k,\mu)$ -nullity distribution_ on a contact metric manifold
$(M^{2n+1},\eta,\xi,\phi,g)$ intro by Blair, Koufogiorgos and Papantoniou [1]
and defined for any $p\in M^{2n+1}$ and $k,\mu\in\mathbb{R}$ as :

$\displaystyle N_{p}(k,\mu)=\\{W\in T_{p}M^{2n+1}:R(U,V)W$ $\displaystyle=$
$\displaystyle k[g(V,W)U-g(U,W)V]$ (1.2)
$\displaystyle+\mu[g(V,W)hU-g(U,W)hV]\\},$

for any $U,V\in T_{p}M$ and $h=\frac{1}{2}\pounds_{\xi}\phi$, where $\pounds$
denotes the Lie differentiation.

In $2009$, Dileo and Pastore [5] introduced a further generalized concept of
the $(k,\mu)$-nullity distribution, called the _$(k,\mu)^{\prime}$ -nullity
distribution_ on an almost Kenmotsu manifold $(M^{2n+1},\eta,\xi,\phi,g)$ and
is defined for any $p\in M^{2n+1}$ and $k,\mu\in\mathbb{R}$ as :

$\displaystyle N_{p}(k,\mu)^{\prime}=\\{W\in T_{p}M^{2n+1}:R(U,V)W$
$\displaystyle=$ $\displaystyle k[g(V,W)U-g(U,W)V]$ (1.3)
$\displaystyle+\mu[g(V,W)h^{\prime}U-g(U,W)h^{\prime}V]\\},$

for any $U,V\in T_{p}M$ and $h^{\prime}=h\circ\phi$.

A Riemannian metric $g$ on a Riemannian manifold $M$ is called a _gradient
$(m,\rho)$-quasi Einstein soliton_ if there exists a smooth function
$f:M^{n}\rightarrow\mathbb{R}$ and three real constants $\rho$, $\lambda$ and
$m$ $(0<m\leq\infty)$ such that

$S+\nabla^{2}f-\frac{1}{m}df\otimes df=\beta g=(\rho r+\lambda)g,$ (1.4)

where $\nabla^{2}$ and $\otimes$ indicate the Hessian of $g$ and tensor
product, respectively. The expression $S+\nabla^{2}f-\frac{1}{m}df\otimes df$
is the $m$-Bakry-Emery Ricci tensor, which is proportional to the metric $g$
and $\lambda=constant$ [20]. The soliton becomes trivial if the potential
function $f$ is constant and the triviality condition implies that the
manifold is an Einstein manifold. Furthermore, when $m=\infty$, the foregoing
equation reduces to gradient $\rho$-Einstein soliton. This notion was
introduced in [2] and recently, Venkatesha et al. studied $\rho$\- Einstein
solitons [12] on almost Kenmotsu manifold. In this connection, the properties
of $(m,\rho)$-quasi Einstein solitons in different geometrical structures have
been studied (in details) by ([6], [8]) and others. To know more about almost
Kenmotsu manifold, here we may mention the work of Venkatesha et al. and Wang
and his collaborator ([13], [15], [17], [19]). Recently, Wang [14] has studied
Yamabe solitons and gradient Yamabe solitons within the context of almost
Kenmotsu $(k,\mu)^{\prime}$ manifolds.

Motivated from the above studies, we make the contribution to investigate
gradient $(m,\rho)$-quasi Einstein solitons in almost Kenmotsu Manifolds.

The current paper is constructed as : In section 2, we recall some basic facts
and formulas of almost Kenmotsu manifolds which we will need throughout the
paper. In sections 3, we characterize gradient $(m,\rho)$-quasi Einstein
solitons with nullity distribution and obtain some interesting results. In the
next section we consider the gradient $(m,\rho)$-quasi Einstein soliton in a
$3$-dimensional Kenmotsu manifold and proved that the manifold is of constant
sectional curvature $-1$, provided $r\neq-(\lambda+2)$. Then we consider an
example to verify the result of our paper.

##  2 Almost Kenmotsu manifolds

In this section we gather the formulas and results of almost Kenmotsu
manifolds which will be required on later sections. A differentiable manifold
$M^{2n+1}$ of dimension $(2n+1)$ is called _almost contact metric manifold_ if
it admits a covariant vector field $\eta$, a contravariant vector field $\xi$,
a $(1,1)$ tensor field $\phi$ and a Riemannian metric $g$ such that

$\phi^{2}=-I+\eta\otimes\xi,\;\eta(\xi)=1,$ (2.1) $g(\phi U,\phi
V)=g(U,V)-\eta(U)\eta(V),$

where $I$ indicates the identity endomorphism . Then also $\phi\xi=0$ and
$\eta\circ\phi=0$; in a straight forward calculation both can be extracted
from (2.1).

On an almost Kenmotsu manifold $M^{2n+1}$, the two symmetric tensor fields
$h=\frac{1}{2}\pounds_{\xi}\phi$ and $l=R(\cdot,\xi)\xi$, satisfy the
following relations [5]

$h\xi=0,\;l\xi=0,\;tr(h)=0,\;tr(h^{\prime})=0,\;h\phi+\phi h=0,$ (2.2)
$\nabla_{U}\xi=-\phi^{2}U+h^{\prime}U(\Rightarrow\nabla_{\xi}\xi=0),$ (2.3)
$\phi l\phi-l=2(h^{2}-\phi^{2}),$ (2.4) $R(U,V)\xi=\eta(U)(V-\phi
hV)-\eta(V)(U-\phi hU)+(\nabla_{V}\phi h)U-(\nabla_{U}\phi h)V,$ (2.5)

for any vector fields $U,V$.

Let the Reeb vector field $\xi$ of an almost Kenmotsu manifold belonging to
the $(k,\mu)^{\prime}$-nullity distribution. Then the symmetric tensor field
$h^{\prime}$ of type $(1,1)$ satisfies the relations $h^{\prime}\phi+\phi
h^{\prime}=0$ and $h^{\prime}\xi=0$. Also, it is understandable that

$h^{\prime 2}=(k+1)\phi^{2}\,(\Leftrightarrow
h^{2}=(k+1)\phi^{2}),\;\;h=0\Leftrightarrow h^{\prime}=0.$ (2.6)

From the equations (2.1) and (2.6), it follows that $k\leq-1$. Certainly, by
(2.6), we conclude that $h^{\prime}$ vanishes if and only if $k=-1$. In view
of the Proposition 4.1 of [5] on a $(k,\mu)^{\prime}$-almost Kenmotsu manifold
with $k<-1$ leads to $\mu=-2$.

For an almost Kenmotsu manifold, we possess from (2.3)

$R(U,V)\xi=k[\eta(V)U-\eta(U)V]+\mu[\eta(V)h^{\prime}U-\eta(U)h^{\prime}V],$
(2.7)
$R(\xi,U)V=k[g(U,V)\xi-\eta(V)U]+\mu[g(h^{\prime}U,V)\xi-\eta(V)h^{\prime}U],$
(2.8)

where $k,\mu\in\mathbb{R}$. Contracting $V$ in (2.8) we have

$S(U,\xi)=2k\eta(U).$ (2.9)

Also, the distribution which is indicated by $\mathcal{D}$ (defined by
$\mathcal{D}=$ker $\eta$). Presume $X\in\mathcal{D}$ be the eigen vector of
$h^{\prime}$ corresponding to the eigen value $\lambda$. Then
$\lambda^{2}=-(k+1)$, a constant, which follows from (2.6). Therefore
$k\leq-1$ and $\lambda=\pm\sqrt{-k-1}$. The non-zero eigen value $\lambda$ and
$-\lambda$ are respectively indicated by $[\lambda]^{\prime}$ and
$[-\lambda]^{\prime}$, which are the corresponding eigen spaces associated
with $h^{\prime}$. Now before introducing the detailed proof of our main
theorem, we first write the following lemma:

######  Lemma 2.1.

(Lemma. 3.2 of [18]) Let $(M^{2n+1},\eta,\xi,\phi,g)$ be an almost Kenmotsu
manifold such that $\xi$ belongs to the $(k,\mu)^{\prime}$-nullity
distribution and $h^{\prime}\neq 0$. Then the Ricci operator $Q$ of $M^{2n+1}$
is given by

$\displaystyle Q=-2nid+2n(k+1)\eta\otimes\xi-2nh^{\prime},$ (2.10)

where $k<-1$, moreover, the scalar curvature of $M^{2n+1}$ is $2n(k-2n)$.

##  3 Gradient $(m,\rho)$-quasi Einstein solitons on a $(2n+1)$-dimensional
almost Kenmotsu manifold

Let us assume that the Riemannian metric of a $(2n+1)$-dimensional almost
Kenmotsu manifold such that $\xi$ belongs to the $(k,\mu)^{\prime}$-nullity
distribution and $h^{\prime}\neq 0$, is a $(m,\rho)$-quasi Einstein soliton.
Then the equation (1.4) may be expressed as

$\nabla_{U}Df+QU=\frac{1}{m}g(U,Df)Df+\beta U.$ (3.1)

Taking covariant derivative of (3.1) along the vector field $V$, we get

$\displaystyle\nabla_{V}\nabla_{U}Df$ $\displaystyle=$
$\displaystyle-\nabla_{V}QU+\frac{1}{m}\nabla_{V}g(U,Df)Df$ (3.2)
$\displaystyle+\frac{1}{m}g(U,Df)\nabla_{V}Df+\beta\nabla_{V}U.$

Interchanging $U$ and $V$ in (3.2), we lead

$\displaystyle\nabla_{U}\nabla_{V}Df$ $\displaystyle=$
$\displaystyle-\nabla_{U}QV+\frac{1}{m}\nabla_{U}g(V,Df)Df$ (3.3)
$\displaystyle+\frac{1}{m}g(V,Df)\nabla_{U}Df+\beta\nabla_{U}V$

and

$\displaystyle\nabla_{[U,V]}Df=-Q[U,V]+\frac{1}{m}g([U,V],Df)Df+\beta[U,V].$
(3.4)

Equations (3.1)-(3.4) and the symmetric property of Levi-Civita connection
together with
$R(U,V)Df=\nabla_{U}\nabla_{V}Df-\nabla_{V}\nabla_{U}Df-\nabla_{[U,V]}Df$ we
lead

$\displaystyle R(U,V)Df$ $\displaystyle=$
$\displaystyle(\nabla_{V}Q)U-(\nabla_{U}Q)V+\frac{\beta}{m}\\{(Vf)U-(Uf)V\\}$
(3.5) $\displaystyle+\frac{1}{m}\\{(Uf)QV-(Vf)QU\\}.$

Taking inner product of (3.5) with $\xi$, we have

$\displaystyle g(R(U,V)Df,\xi)$ $\displaystyle=$
$\displaystyle\frac{\beta}{m}\\{(Vf)\eta(U)-(Uf)\eta(V)\\}$ (3.6)
$\displaystyle+\frac{1}{m}\\{(Uf)\eta(QV)-(Vf)\eta(QU)\\}.$

Again (2.7) implies that

$g(R(U,V)\xi,Df)=k[\eta(V)(Uf)-\eta(U)(Vf)]+\mu[\eta(V)(h^{\prime}Uf)-\eta(U)(h^{\prime}Vf)].$
(3.7)

Combining equation (3.6) and (3.7) reveal that

$\displaystyle
k[\eta(U)(Vf)-\eta(V)(Uf)]+\mu[\eta(U)(h^{\prime}Vf)-\eta(V)(h^{\prime}Uf)]$
$\displaystyle=\frac{\beta}{m}\\{(Vf)\eta(U)-(Uf)\eta(V)\\}$
$\displaystyle+\frac{1}{m}\\{(Uf)\eta(QV)-(Vf)\eta(QU)\\}.$ (3.8)

Setting $V=\xi$ in the foregoing equation yields

$2mh^{\prime}Df=(\beta-2nk-mk)\\{Df-(\xi f)\xi\\},$ (3.9)

where we have used $\mu=-2$.

Letting $\beta=(2nk+mk+2m)$ and taking into account the equation (2.6) and
operating $h^{\prime}$ on (3.9) produces that

$-(k+1)\\{Df-(\xi f)\xi\\}=h^{\prime}Df,$ (3.10)

Comparing the antecedent relation with (3.9) gives that

$(k+2)\\{Df-(\xi f)\xi\\}=0.$ (3.11)

This shows that either $k=-2$ or $\\{Df-(\xi f)\xi\\}=0$.

Case i: If $k=-2$, then the Proposition 4.1 and Corollary 4.2 of [5] state
that $M^{2n+1}$ is locally isometric to the Riemannian product
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. In fact, from ([9],[10]) we can
say that the product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ is a rigid
gradient Ricci soliton.

Case ii:

$Df=(\xi f)\xi.$ (3.12)

Executing the covariant differentiation of (3.12) along $U\in\chi(M)$ and
utilizing (2.3) we get

$\nabla_{U}Df=U(\xi f)\xi+(\xi f)U-(\xi f)\eta(U)\xi+(\xi f)h^{\prime}U.$
(3.13)

Replacing the foregoing equation into (3.1) revels that

$QX=\beta U-(\xi f)U-U(\xi f)\xi+(\xi f)\eta(U)\xi-(\xi
f)h^{\prime}U+\frac{1}{m}g(U,Df)Df.$ (3.14)

Comparing (2.10) and (3.14) give that

$\displaystyle\\{\beta+2n-(\xi f)\\}U+\\{2n-(\xi f)\\}h^{\prime}U+\\{(\xi
f)\eta(U)$ $\displaystyle-U(\xi
f)-2n(k+1)\eta(U)\\}\xi+\frac{1}{m}g(U,Df)Df=0.$ (3.15)

Now operating $h^{\prime}$ we get

$\displaystyle\\{\beta+2n-(\xi f)\\}h^{\prime}U+(k+1)\\{2n-(\xi
f)\\}\phi^{2}U$ $\displaystyle+\frac{1}{m}g(U,Df)h^{\prime}Df=0.$ (3.16)

Contracting $U$ in the above equation we infer

$2n(k+1)\\{2n-(\xi f)\\}+\frac{1}{m}g(h^{\prime}Df,Df)=0.$ (3.17)

Putting $U=\xi$ in (3.16), we have

$\eta(Df)h^{\prime}Df=0.$ (3.18)

This shows that either $\eta(Df)=0$ or $h^{\prime}Df=0.$

Case (i): If $\eta(Df)=0$, then we get $\xi f=0$ and hence from equation
(3.12) it follows that $f=constant$. Then we get from (3.1) that the manifold
is an Einstein manifold.

Case (ii): If $h^{\prime}Df=0$, then from (3.17) we get $(\xi f)=2n$. Now
putting this value in (3.16) we obtain $\beta h^{\prime}U=0$. Hence either
$\beta=0$, or $h^{\prime}U=0$. Both contradicts our assumptions
$\beta=(2n+m)k+2m$ and $k<-1$ respectively .

Thus, we can state the following theorem:

######  Theorem 3.1.

Let the Riemannian metric of a $(2n+1)$-dimensional $(k,\mu)^{\prime}$ almost
Kenmotsu manifold with $h^{\prime}\neq 0$, be the gradient $(m,\rho)$-quasi
Einstein soliton. Then either $M^{2n+1}$ is locally isometric to a rigid
gradient Ricci soliton $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or
$M^{2n+1}$ is an Einstein manifold, provided $\beta=(2nk+mk+2m)$.

We know that when $m=\infty$, the $(m,\rho)$-quasi Einstein soliton becomes
gradient $\rho$-Einstein soliton. Putting the value $m=\infty$ in (3) and by a
straight forward calculation we find that the manifold $M^{2n+1}$ is locally
isometric to a rigid gradient Ricci soliton
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. Thus, we can state:

######  Corollary 3.2.

Let the Riemannian metric of a $(2n+1)$-dimensional almost Kenmotsu manifold
such that $\xi$ belongs to the $(k,\mu)^{\prime}$-nullity distribution and
$h^{\prime}\neq 0$, be the gradient $\rho$\- Einstein soliton. Then $M^{2n+1}$
is locally isometric to a rigid gradient Ricci soliton
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$.

######  Remark 3.3.

The above corollary have been proved by Venkatesha and Kumara in their paper
[12]. They also prove that the potential vector field is tangential to the
Euclidean factor $\emph{R}^{n}$.

##  4 Gradient $(m,\rho)$-quasi Einstein solitons on a $3$-dimensional
Kenmotsu manifold

In [5], Dileo and Pastore proved that the following conditions are equivalent:

(1) the (1,1)-type tensor field $h$ vanishes and foliations of the
distribution $\mathcal{D}$ are K$\ddot{a}$hlerian.

(2) almost contact metric structure of an almost Kenmotsu manifold is normal.

As a outcome, we get instantly that a 3-dimensional almost Kenmotsu manifold
reduces to a Kenmotsu manifold if and only if $h=0$. In this section, we
target to investigate the gradient $(m,\rho)$-quasi Einstein soliton on a
$3$-dimensional Kenmotsu manifold. Making use of $h=0$ in equation (2.3) we
get $\nabla\xi=-\phi^{2}$, and this revels that

$R(U,V)\xi=-\eta(V)U+\eta(U)V$ (4.1)

for any $U,V$ $\in\chi(M)$ and therefore by contracting $V$ in (4.1) we obtain
$Q\xi=-2\xi$. From [4] we know that for a 3-dimensional Kenmotsu manifold

$\displaystyle R(U,V)W$ $\displaystyle=$
$\displaystyle(\frac{r+4}{2})[g(V,W)U-g(U,W)V]$
$\displaystyle-(\frac{r+6}{2})[g(V,W)\eta(U)\xi-g(U,W)\eta(V)\xi$
$\displaystyle+\eta(V)\eta(W)U-\eta(U)\eta(W)V],$
$S(U,V)=\frac{1}{2}[(r+2)g(U,V)-(r+6)\eta(U)\eta(V)].$ (4.3)

Now before introducing the detailed proof of our main theorem, we first state
the following result:

######  Lemma 4.1.

(lemma. 4.1 of [16]) For a 3-dimensional Kenmotsu manifold
$(M^{3},\phi,\xi,\eta,g)$, we have

$\displaystyle\xi r=-2(r+6)$ (4.4)

where $r$ denotes the scalar curvature of $M$.

Let us assume that the Riemannian metric of a $3$-dimensional almost Kenmotsu
manifold be a $(m,\rho)$-quasi Einstein soliton. Then the equation (1.4) may
be expressed as

$\nabla_{U}Df+QU=\frac{1}{m}g(U,Df)Df+\beta U.$ (4.5)

Taking covariant derivative of (4.5) along the vector field $V$, we get

$\displaystyle\nabla_{V}\nabla_{U}Df$ $\displaystyle=$
$\displaystyle-\nabla_{V}QU+\frac{1}{m}\nabla_{V}g(U,Df)Df$ (4.6)
$\displaystyle+\frac{1}{m}g(U,Df)\nabla_{V}Df+\beta\nabla_{V}U+(V\beta)U.$

Interchanging $U$ and $V$ in (4.6), we lead

$\displaystyle\nabla_{U}\nabla_{V}Df$ $\displaystyle=$
$\displaystyle-\nabla_{U}QV+\frac{1}{m}\nabla_{U}g(V,Df)Df$ (4.7)
$\displaystyle+\frac{1}{m}g(V,Df)\nabla_{U}Df+\beta\nabla_{U}V+(U\beta)V$

and

$\displaystyle\nabla_{[U,V]}Df=-Q[U,V]+\frac{1}{m}g([U,V],Df)Df+\beta[U,V].$
(4.8)

Equations (4.5)-(4.8) and the symmetric property of Levi-Civita connection
together with
$R(U,V)Df=\nabla_{U}\nabla_{V}Df-\nabla_{V}\nabla_{U}Df-\nabla_{[U,V]}Df$ we
lead

$\displaystyle R(U,V)Df$ $\displaystyle=$
$\displaystyle(\nabla_{V}Q)U-(\nabla_{U}Q)V+\frac{\beta}{m}\\{(Vf)U-(Uf)V\\}$
(4.9)
$\displaystyle+\frac{1}{m}\\{(Uf)QV-(Vf)QU\\}+\\{(U\beta)V-(V\beta)U\\}.$

Taking inner product of (4.9) with $\xi$ and using (4.3) we have

$\displaystyle g(R(U,V)Df,\xi)$ $\displaystyle=$
$\displaystyle\frac{\beta}{m}\\{(Vf)\eta(U)-(Uf)\eta(V)\\}$ (4.10)
$\displaystyle+\frac{1}{m}\\{(Uf)\eta(QV)-(Vf)\eta(QU)\\}$
$\displaystyle+\\{(U\beta)\eta(V)-(V\beta)\eta(U)\\}.$

Again, from (4.1) we infer that

$g(R(U,V)Df,\xi)=-\\{(Vf)\eta(U)-(Uf)\eta(V)\\}.$ (4.11)

Combining equation (4.10) and (4.11) reveal that

$\displaystyle-\\{(Vf)\eta(U)-(Uf)\eta(V)\\}$ $\displaystyle=$
$\displaystyle\frac{\beta}{m}\\{(Vf)\eta(U)-(Uf)\eta(V)\\}$ (4.12)
$\displaystyle+\frac{1}{m}\\{(Uf)\eta(QV)-(Vf)\eta(QU)\\}$
$\displaystyle+\\{(U\beta)\eta(V)-(V\beta)\eta(U)\\}.$

Replacing $V$ by $\xi$ in the foregoing equation, we get

$d(f-\beta)=\xi(f-\beta)\eta,$ (4.13)

where $d$ stands for the exterior differentiation, provided $\beta=-2$. In
other word, $f-\beta$ is invariant along $\mathcal{D}$, i.e., $X(f-\beta)=0$
for any $X\in\mathcal{D}$. Taking into account the above fact and using Lemma
6.1, we infer

$(\xi f)=(\xi\beta)=\rho(\xi r)=-2\rho(r+6).$ (4.14)

The contraction of the equation (4.9) along $U$ and applying Lemma 6.1, we get

$\displaystyle S(V,Df)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Vr)+\frac{2\beta}{m}(Vf)$ (4.15)
$\displaystyle-\frac{1}{m}\\{r(Vf)-g(QV,Df)\\}-2(V\beta).$

Clearly, comparing the above equation with (4.3) yields

$\displaystyle-(Vr)-\frac{4\beta}{m}(Vf)+\frac{2}{m}\\{r(Vf)-g(QV,Df)\\}$
$\displaystyle+4(V\beta)+(r+2)(Vf)-(r+6)\eta(V)\xi f=0.$ (4.16)

By a straight forward calculation, replacing $V$ by $\xi$ in (4) and using
(4.14), we can easily obtain

$(\xi f)=\frac{2m(r+6)}{4\beta-2r-4}.$ (4.17)

Comparing the antecedent equation with (4.14) reveals that

$2(r+6)\\{\rho+\frac{m}{4\beta-2r-4}\\}=0.$ (4.18)

This shows that either $r=-6$ or $\\{\rho+\frac{m}{4\beta-2r-4}\\}=0$. Next,
we split our investigation as :

Case (i): If $r=-6$, then from equation (4.3) we conclude that $S=-2g$.
Therefore, by using equation (4) we sum up that the manifold is of constant
sectional curvature $-1$.

Case (ii): If $\\{\rho+\frac{m}{4\beta-2r-4}\\}=0$, then by a simple
calculation we get

$r=\frac{4\lambda-4-4\rho\lambda-4\rho+m}{2(2\rho^{2}-3\rho+1)}=constant,$
(4.19)

provided $\rho\neq 1$. Hence by applying Lemma 6.1 we can easily get $r=-6$.
Therefore, from Case (i) we see that the manifold is of constant sectional
curvature $-1$. After combining the two conditions namely, $\beta=-2$ and
$\rho\neq 1$, we can write $r\neq-(\lambda+2)$. Thus we have the following
theorem:

######  Theorem 4.2.

Let the Riemannian metric of a $3$-dimensional Kenmotsu manifold be the
gradient $(m,\rho)$-quasi Einstein soliton. Then the manifold is of constant
sectional curvature $-1$, provided $r\neq-(\lambda+2)$.

We know that when $m=\infty$, the $(m,\rho)$-quasi Einstein soliton gives the
so called gradient $\rho$-Einstein soliton. Putting the value $m=\infty$ in
(4.12) and by a straight forward calculation we find that the manifold is of
constant sectional curvature $-1$. Thus, we can state:

######  Corollary 4.3.

Let the Riemannian metric of a $3$-dimensional Kenmotsu manifold be the
gradient $\rho$\- Einstein soliton. Then the manifold is of constant sectional
curvature $-1$.

##  5 Example

Here we consider an example cited in our paper [3].

We consider the 3-dimensional manifold
$M=\\{(u,v,w)\varepsilon\mathbb{R}^{3},w\neq 0\\},$ where $(u,v,w)$ are
standard coordinate of $\mathbb{R}^{3}.$

The vector fields

$E_{1}=w\frac{\partial}{\partial u},\hskip
7.0ptE_{2}=w\frac{\partial}{\partial v},\hskip
7.0ptE_{3}=-w\frac{\partial}{\partial w}$

are linearly independent at each point of $M.$

Let $g$ be the Riemannian metric defined by

$g(E_{1},E_{3})=g(E_{1},E_{2})=g(E_{2},E_{3})=0,$
$g(E_{1},E_{1})=g(E_{2},E_{2})=g(E_{3},E_{3})=1.$

Let $\eta$ be the 1-form defined by $\eta(W)=g(W,E_{3})$ for any
$W\varepsilon\chi(M).$

Let $\phi$ be the $(1,1)$ tensor field defined by

$\phi(E_{1})=-E_{2},\hskip 7.0pt\phi(E_{2})=E_{1},\hskip 7.0pt\phi(E_{3})=0.$

Then for $E_{3}=\xi$ , the structure $(\phi,\xi,\eta,g)$ defines an almost
contact metric structure on $M$. Further Koszul’s formula yields

$\nabla_{E_{1}}E_{3}=E_{1},\hskip 10.0pt\nabla_{E_{1}}E_{2}=0,\hskip
10.0pt\nabla_{E_{1}}E_{1}=-E_{3},$ $\nabla_{E_{2}}E_{3}=E_{2},\hskip
10.0pt\nabla_{E_{2}}E_{2}=E_{3},\hskip 10.0pt\nabla_{E_{2}}E_{1}=0,$
$\nabla_{E_{3}}E_{3}=0,\hskip 10.0pt\nabla_{E_{3}}E_{2}=0,\hskip
10.0pt\nabla_{E_{3}}E_{1}=0.$ (5.1)

From the above it follows that the manifold satisfies
$\nabla_{U}\xi=U-\eta(U)\xi$, for $\xi=E_{3}$. Hence the manifold is a
Kenmotsu manifold.
We verified that

$R(E_{1},E_{2})E_{3}=0,\hskip 10.0ptR(E_{2},E_{3})E_{3}=-E_{2},\hskip
10.0ptR(E_{1},E_{3})E_{3}=-E_{1},$ $R(E_{1},E_{2})E_{2}=-E_{1},\hskip
10.0ptR(E_{2},E_{3})E_{2}=E_{3},\hskip 10.0ptR(E_{1},E_{3})E_{2}=0,$
$R(E_{1},E_{2})E_{1}=E_{2},\hskip 10.0ptR(E_{2},E_{3})E_{1}=0,\hskip
10.0ptR(E_{1},E_{3})E_{1}=E_{3}.$

From the above expressions of the curvature tensor $R$ we obtain

$\displaystyle S(E_{1},E_{1})$ $\displaystyle=$ $\displaystyle
g(R(E_{1},E_{2})E_{2},E_{1})+g(R(E_{1},E_{3})E_{3},E_{1})$ (5.2)
$\displaystyle=$ $\displaystyle-2.$

Similarly, we have

$S(E_{2},E_{2})=S(E_{3},E_{3})=-2.$

Therefore,

$r=S(E_{1},E_{1})+S(E_{2},E_{2})+S(E_{3},E_{3})=-6.$

Let $f:M^{3}\rightarrow\mathbb{R}$ be a smooth function defined by $f=-w^{2}$.
Then gradient of $f$ with respect to $g$ is given by

$Df=-2w\frac{\partial}{\partial w}=2E_{3}.$

With the help of (5.1) we can easily get

$Hessf(E_{3},E_{3})=0.$

Thus gradient $\rho$-Einstein soliton equation shows

$Hessf(E_{3},E_{3})+S(E_{3},E_{3})+2g(E_{3},E_{3})=0.$

Similarly checking the other components we conclude that that $M^{3}$
satisfies

$Hessf(U,V)+S(U,V)+2g(U,V)=0.$

Thus $g$ is a gradient $\rho$-Einstein soliton with $f=-w^{2}$ and $\beta=-2$.
Hence the Corollary 4.3 is verified.

##  6 Declarations

###  6.1 Funding

Not applicable.

###  6.2 Conflicts of interest/Competing interests

The authors declare that they have no conflict of interest.

###  6.3 Availability of data and material

Not applicable.

###  6.4 Code availability

Not applicable.

## References

  * [1] Blair, D.E., Koufogiorgos, T. and Papantoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel. J. Math. 91(1995), 189-214.
  * [2] Catino, G. and Mazzieri, L., Gradient Einstein-solitons, arXiv:1201.6620.
  * [3] De, U. C. and De, K., On $\phi$-Concircularly Symmetric Kenmotsu manifolds, Thai Journal of Mathematics 10 (2012), 1-11.
  * [4] De, U.C. and Pathak, G., On 3-dimensional Kenmotsu manifold, Indian J. Pure Appl. Math., 35(2004), 159-165.
  * [5] Dileo, G. and Pastore, A.M., Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93(2009), 46-61.
  * [6] Ghosh, A., $(m,\rho)$-quasi-Einstein metric within the framework of K-contact manifolds, J. Math. Phys. Anal. Geom. 17 (2014), 369-376.
  * [7] Gray, A., Spaces of constancy of curvature operators, Proc. Amer. Math. Soc., 17(1966), 897-902.
  * [8] Huang, G. and Wei, Y., The classification of $(m,\rho)$-quasi-Einstein manifolds, Ann. Global Anal. Geom., 44 (2013), 269-282.
  * [9] Petersen, P. and Wylie, W., Rigidity of gradient Ricci solitons, Pacific J. Math. 241(2009), 329-345.
  * [10] Petersen, P. and Wylie, W., On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137(2009), 2085-2092.
  * [11] Tanno, S., Some differential equations on Riemannian manifolds, J. Math. Soc. Japan, 30(1978), 509-531.
  * [12] Venkatesha, V. and Kumara, H. A., gradient $\rho$-Einstein soliton on almost Kenmotsu manifolds, ANNALI DELL’UNIVERSITA’ DI FERRARA 65(2019), 375-388.
  * [13] Venkatesha, V., Naik, D. M. and Kumara, H. A., $\ast$-Ricci solitons and gradient almost $\ast$-Ricci solitons on Kenmotsu manifolds, Math. Slovaca 69 (2019), 1447-1458.
  * [14] Wang, Y., Almost Kenmotsu $(k,\mu)^{\prime}$-manifolds with Yamabe solitons , RACSAM 115, 14 (2021). https://doi.org/10.1007/s13398-020-00951-y
  * [15] Wang, Y., Conformally flat almost Kenmotsu 3-manifolds, Mediterr. J. Math. 14 (2017), DOI https://doi.org/10.1007/s00009-017-0984-9.
  * [16] Wang, Y., Gradient Ricci almost Solitons on two classes of almost Kenmotsu Manifolds, J. Korean Math. Soc. 23 (2016), 1101-1114.
  * [17] Wang, Y. and Dai, X., Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds, Inter. J. Geom. Meth. Modern Phys. 16 (2019), 1950092.
  * [18] Wang, Y. and Liu, X., On almost Kenmotsu manifolds satisfying some nullity distributions, Proc. Natl. Acad. Sci., India,Sect. A Phys. Sci 86(2016), 347-353.
  * [19] Wang, Y. and Liu, X., A Schur-type theorem for CR-integrable almost Kenmotsu manifolds, Math. Slovaca 66 (2016), 1217-1226.
  * [20] Wei, G. and Wylie, W., Comparison geometry for the Bakry-Emery Ricci tensor, Journal of Differential Geometry, 83(2009), 337-405. doi:10.4310/jdg/1261495336.