<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta>
      <journal-title-group>
        <journal-title>SEBD</journal-title>
      </journal-title-group>
    </journal-meta>
    <article-meta>
      <title-group>
        <article-title>Scaling Bio-Inspired Neural Features to Real-World Image Retrieval Problems</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Gabriele Lagani</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>University of Pisa</institution>
          ,
          <addr-line>56127, Pisa</addr-line>
          ,
          <country country="IT">Italy</country>
        </aff>
      </contrib-group>
      <pub-date>
        <year>2023</year>
      </pub-date>
      <volume>31</volume>
      <fpage>02</fpage>
      <lpage>05</lpage>
      <abstract>
        <p>In the last decade, approaches in feature extraction for content-based multimedia retrieval exploited neural feature representations to describe complex data types such as images. In particular, recent approaches proposed to leverage bio-inspired learning solutions, which have the advantage to ofer better generalization from fewer training samples. However, scaling these solutions to real-world datasets is a challenging problem. In my recent research, I proposed a possible approach to achieve such scalability, based on translating bio-inspired learning models into matrix multiplications, which can eficiently be executed on GPU. In this way, for the first time, I was able to validate bio-inspired methodologies on large-scale datasets such as ImageNet.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;Hebbian Learning</kwd>
        <kwd>Deep Learning</kwd>
        <kwd>Bio-Inspired</kwd>
        <kwd>Neural Networks</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>
        Interpreting and retrieving multimedia data is dificult due to the high level of semantic
abstraction with which information is represented. Deep Learning (DL) provides a valid aid
to handle this type of information. For example, in the context of image data, Deep Neural
Networks (DNNs), trained on supervised object recognition tasks, can provide highly abstract
feature representations, which are useful for indexing and Content-Based Image Retrieval (CBIR)
[
        <xref ref-type="bibr" rid="ref1 ref2 ref3">1, 2, 3</xref>
        ].
      </p>
      <p>
        A pitfall of supervised training is the requirement of large amounts of labeled data, which are
typically dificult to gather, as opposed to unlabeled data. In this light, previous work focused on
semi-supervised learning approaches, by which labeled data could be enhanced by large amount
of unlabeled data, in order to learn higher quality feature extractors with fewer labels [
        <xref ref-type="bibr" rid="ref4 ref5 ref6 ref7">4, 5, 6, 7</xref>
        ].
      </p>
      <p>
        Recently, it was shown that bio-inspired unsupervised learning solutions based on the Hebbian
principle [
        <xref ref-type="bibr" rid="ref10 ref8 ref9">8, 9, 10</xref>
        ] were able to achieve higher results than traditional counterparts such as
Variational Auto-Encoder (VAE) based pre-training [
        <xref ref-type="bibr" rid="ref11 ref6">11, 6</xref>
        ], especially in learning regimes with
very scarce label availability. Hebbian learning rules provide local mechanisms for synaptic
adaptation, which are connected to data analysis operations such as clustering or principal
component analysis [
        <xref ref-type="bibr" rid="ref12">12</xref>
        ], that enable training layers independently from the next ones, allowing
a more efective exploitation of the available training set information. However, the application
of such approaches has remained limited to simple datasets like CIFAR-10 and CIFAR-100 [13].
      </p>
      <p>As part of my PhD work, I aimed at extending the analysis of such methodologies also to
more realistic datasets such as ImageNet [14]. This is a collection of over 1.2 million
realworld images crawled from the web, categorized into 1000 distinct classes. The problem of
Hebbian approaches in such scenarios is that they do not scale well with the complexity of the
problem at hand. To overcome this limitation, I developed FastHebb [15], an eficient solution
for Hebbian feature learning and extraction that is based on re-expressing bio-inspired Hebbian
synaptic equations in terms of matrix multiplications, which can be executed very eficiently by
leveraging GPU acceleration. This solution permitted to achieve a speedup of Hebbian training
up to 50x compared to other solutions, while producing feature representations that proved
to be very efective in the context of neural feature-based CBIR, when validated on various
benchmarks, including ImageNet.</p>
      <p>The remainder of this document gives an overview of the developments mentioned above,
organized according to the following structure: Section 2 discusses some related and background
material; Section 3 describes more in detail bio-inspired Hebbian feature learning for retrieval,
and, in particular, the FastHebb solution; Section 4 provides some empirical results to validate
the proposed approaches; Finally, Section 5 presents some concluding remarks.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Related Work</title>
      <p>
        Multimedia content retrieval, and, in particular, CBIR, has observed great benefits from
transitioning from handcrafted feature representations learned ones. This is due to the semantic gap
between these types of features: learned representations can encode highly abstract concepts
[
        <xref ref-type="bibr" rid="ref1">1</xref>
        ]. Therefore, previous work has shown experimental evidence for the efectivenes of such
features in image retrieval tasks [16, 17, 18, 19]. While these methods use features obtained from
DNNs pre-trained on image classification tasks, in [ 20] the authors proposed an end-to-end
training procedure specifically designed for CBIR. They used a siamese architecture with a
triplet loss that pushes related images close in feature space (according to a given ground-truth),
while pushing unrelated images away. Finally, Bai et al. [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ] presented a comprehensive
experimental comparison of various methods on modern computer vision datasets, including their
proposed Optimized AlexNet for Image Retrieval (OANIR) approach, in which they applied an
AlexNet-inspired [21] network architecture specifically modified and optimized for the retrieval
task.
      </p>
      <p>
        One of the challenges of retrieval tasks is that datasets have typically a very large scale.
Manually providing ground-truth labels for training might become very expensive in these
cases, suggesting that semi-supervised learning techniques might be exploited in these cases
[
        <xref ref-type="bibr" rid="ref4 ref5 ref6 ref7">5, 4, 6, 7</xref>
        ]. Recently, bio-inspired Hebbian learning approaches have shown great promises for
unsupervised [
        <xref ref-type="bibr" rid="ref12">22, 23, 24, 25, 26, 12</xref>
        ] semi-supervised learning [
        <xref ref-type="bibr" rid="ref10 ref8 ref9">8, 9, 10</xref>
        ], but their capability to
scale to large datasets has remained limited.
      </p>
      <p>In this contribution, I describe a recent solution, named FastHebb, that I developed as part of
my PhD, which enabled scaling Hebbian-based solutions for image recognition and retrieval
also to real-world and large scale datasets such as ImageNet[14].</p>
    </sec>
    <sec id="sec-3">
      <title>3. Scalable Neural Features from Bio-Inpired Learning: FastHebb</title>
      <p>
        I have explored diferent types of bio-inspired learning rules, but in the following I will focus
on two types in particular: Hebbian Principal Component Analysis (HPCA) and
Soft-WinnerTakes-All (STWA). I will not delve deep into the theoretical details of these approaches, but the
interested reader can refer to [
        <xref ref-type="bibr" rid="ref12">27, 26, 12</xref>
        ].
      </p>
      <p>These learning rules follow a local learning scheme, by which a neuron updates its weights
based on information that is available at the neuron site, according to biological constraints.
This is opposed to traditional backprop learning where an end-to-end error delivery mechanism
takes place. However, when dealing with convolutional network layers, neurons at diferent
horizontal/vertical ofsets need to maintain shared weights, so they must undergo the same
weight updates. This is achieved by aggregating weight updates at diferent ofsets through a
(weighted) averaging mechanism (Fig. 1).</p>
      <p>This subdivision into phases of update computation and aggregation results in slow processing
when dealing with large data streams. In order to overcome this limitation, the ideas of FastHebb
are twofold: first, update computation and aggregation phases are merged in a unique phase;
second, Hebbian update computations are translated into matrix multiplications. The resulting
computation can leverage GPU processing much more eficiently, enabling Hebbian-based
approaches at a much larger scale. The details can be found in [15].</p>
    </sec>
    <sec id="sec-4">
      <title>4. Experiments and Results</title>
      <p>
        This Section presents an experimental validation of the FastHebb method in the context of
CBIR. In order to show the scalability of the FastHebb approach to complex scenarios, the
ImageNet ILSVRC2021 [14] benchmark was used for evaluation. To evaluate the approaches
in conditions of label scarcity, I considered various sample eficiency regimes, i.e. scenarios in
which we assume that only a given percentage of the available training samples is labeled. We
adopted a semi-supervised training protocol where a network model (shown in Fig. 2) was
pre-trained using Hebbian methods on all the available training samples, and then fine-tuned
using supervised backprop training on the labeled samples only. As a baseline for comparison,
we considered Variational Auto-Encoder (VAE) based semi-supervised training [
        <xref ref-type="bibr" rid="ref6">6</xref>
        ].
      </p>
      <p>In order to assess the improvements in scaling properties of FastHebb, compared to traditional
Hebbian learning, we measured duration and total number of training epochs required by each
approach, for the unsupervised pre-training phase (while the successive supervised fine tuning
phase is comparable across methods). The number of epochs is counted as the epoch at which
weights stop changing. The results, shown in Tab. 1 , demonstrate significant improvements (up
to 50x) due to the FastHebb solution. Moreover, when the number of epochs is also considered,
Hebbian pre-training results to be faster than the backprop-based VAE counterpart.</p>
      <p>Trained networks were then used to extract feature representations from the last hidden layer
(before the final classifier), to be used for retrieval. The neural feature based retrieval process
works as follows: given a query image, taken from the test set, its feature representation is
computed by feeding it to the network model. The feature representation of dataset images is
also pre-computed in the same way, and stored for indexing and retrieval purposes. Given the
query feature representation, we search for the closest dataset elements in feature space, and
rank them according to Euclidean distance. Finally, we evaluate the mean Average Precision
(mAP) over all the queries, where a retrieved image is considered a positive if its label, according
to the ground-truth, corresponds to the query label.</p>
      <p>Tab. 2 shows the retrieval results, in terms of mAP, achieved with the diferent methods, in
various regimes of sample eficiency. It should be noticed that the results without FastHebb are
not reported in this Table, because they coincide with the FastHebb results. Indeed, FastHebb
is a reformulation of weight updates that improves the computational eficiency, but it does
not change the synaptic dynamics itself. From the results we can see that Hebbian approaches
exhibit better results than the backprop-based VAE method, especially in regimes with fewer
labels (below 10%). The latter is able to improve only when a larger number of labels is available
for the fine-tuning phase.</p>
    </sec>
    <sec id="sec-5">
      <title>5. Conclusions</title>
      <p>The problem of sample eficient training for DNNs is of strong practical interest, due to the
dificulty to gather labeled training samples. In this contribution, I discussed some
semisupervised training approaches based on bio-inspired Hebbian learning methods, which are
promising in these scenarios. Scaling these approaches to real-world CBIR settings, which are
well captured by benchmarks such as ImageNet, is a significant challenge, that I proposed to
address through the FastHebb solution. Results showed a significant performance increase
during training thanks to FastHebb. Moreover, evaluation of Hebbian neural features in retrieval
settings showed promising results, especially in scenarios of label scarcity.</p>
      <p>As possible future work directions, I suggest to explore further Hebbian rules which can be
used for feature extraction and unsupervised pre-training, for example derived from Independent
Component Analysis (ICA) [28]. Finally, in the context of semi-supervised learning, Hebbian
approaches can also be combined with pseudo-labeling and consistency-based methods [29, 30].</p>
    </sec>
    <sec id="sec-6">
      <title>Acknowledgments</title>
      <p>This work was partially supported by:
- Tuscany Health Ecosystem (THE) Project (CUP I53C22000780001), funded by the National
Recovery and Resilience Plan (NRPP), within the NextGeneration Europe (NGEU) Program;
- AI4Media project, funded by the EC (H2020 - Contract n. 951911);
- INAROS (INtelligenza ARtificiale per il mOnitoraggio e Supporto agli anziani) project co-funded
by Tuscany Region POR FSE CUP B53D21008060008.
[13] A. Krizhevsky, G. Hinton, Learning multiple layers of features from tiny images, 2009.
[14] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, L. Fei-Fei, Imagenet: A large-scale hierarchical
image database, in: 2009 IEEE conference on computer vision and pattern recognition,
Ieee, 2009, pp. 248–255.
[15] G. Lagani, C. Gennaro, H. Fassold, G. Amato, Fasthebb: Scaling hebbian training of deep
neural networks to imagenet level, in: International Conference on Similarity Search and
Applications, Springer, 2022, pp. 251–264.
[16] A. Babenko, A. Slesarev, A. Chigorin, V. Lempitsky, Neural codes for image retrieval, in:</p>
      <p>European conference on computer vision, Springer, 2014, pp. 584–599.
[17] A. Babenko, V. Lempitsky, Aggregating deep convolutional features for image retrieval,
arXiv preprint arXiv:1510.07493 (2015).
[18] J. Yue-Hei Ng, F. Yang, L. S. Davis, Exploiting local features from deep networks for
image retrieval, in: Proceedings of the IEEE conference on computer vision and pattern
recognition workshops, 2015, pp. 53–61.
[19] A. Gordo, J. Almazán, J. Revaud, D. Larlus, Deep image retrieval: Learning global
representations for image search, in: European conference on computer vision, Springer, 2016,
pp. 241–257.
[20] A. Gordo, J. Almazan, J. Revaud, D. Larlus, End-to-end learning of deep visual
representations for image retrieval, International Journal of Computer Vision 124 (2017) 237–254.
[21] A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional
neural networks, Advances in neural information processing systems 25 (2012) 1097–1105.
[22] A. Wadhwa, U. Madhow, Bottom-up deep learning using the hebbian principle, 2016.
[23] Y. Bahroun, A. Soltoggio, Online representation learning with single and multi-layer
hebbian networks for image classification, in: International Conference on Artificial
Neural Networks, Springer, 2017, pp. 354–363.
[24] T. Miconi, Multi-layer hebbian networks with modern deep learning frameworks, arXiv
preprint arXiv:2107.01729 (2021).
[25] M. Gupta, S. K. Modi, H. Zhang, J. H. Lee, J. H. Lim, Is bio-inspired learning better than
backprop? benchmarking bio learning vs. backprop, arXiv preprint arXiv:2212.04614
(2022).
[26] G. Lagani, F. Falchi, C. Gennaro, G. Amato, Training convolutional neural networks
with competitive hebbian learning approaches, in: International Conference on Machine
Learning, Optimization, and Data Science, Springer, 2021, pp. 25–40.
[27] G. Amato, F. Carrara, F. Falchi, C. Gennaro, G. Lagani, Hebbian learning meets deep
convolutional neural networks, in: International Conference on Image Analysis and
Processing, Springer, 2019, pp. 324–334.
[28] A. Hyvarinen, J. Karhunen, E. Oja, Independent component analysis, Studies in informatics
and control 11 (2002) 205–207.
[29] A. Iscen, G. Tolias, Y. Avrithis, O. Chum, Label propagation for deep semi-supervised
learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
Recognition, 2019, pp. 5070–5079.
[30] P. Sellars, A. I. Aviles-Rivero, C.-B. Schönlieb, Laplacenet: A hybrid energy-neural model
for deep semi-supervised classification, arXiv preprint arXiv:2106.04527 (2021).</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <ref id="ref1">
        <mixed-citation>
          [1]
          <string-name>
            <given-names>J.</given-names>
            <surname>Wan</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Wang</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S. C. H.</given-names>
            <surname>Hoi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Wu</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Zhu</surname>
          </string-name>
          ,
          <string-name>
            <given-names>Y.</given-names>
            <surname>Zhang</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Li</surname>
          </string-name>
          ,
          <article-title>Deep learning for content-based image retrieval: A comprehensive study</article-title>
          ,
          <source>in: Proceedings of the 22nd ACM international conference on Multimedia</source>
          ,
          <year>2014</year>
          , pp.
          <fpage>157</fpage>
          -
          <lpage>166</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref2">
        <mixed-citation>
          [2]
          <string-name>
            <given-names>C.</given-names>
            <surname>Bai</surname>
          </string-name>
          ,
          <string-name>
            <given-names>L.</given-names>
            <surname>Huang</surname>
          </string-name>
          ,
          <string-name>
            <given-names>X.</given-names>
            <surname>Pan</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Zheng</surname>
          </string-name>
          , S. Chen,
          <article-title>Optimization of deep convolutional neural network for large scale image retrieval</article-title>
          ,
          <source>Neurocomputing</source>
          <volume>303</volume>
          (
          <year>2018</year>
          )
          <fpage>60</fpage>
          -
          <lpage>67</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref3">
        <mixed-citation>
          [3]
          <string-name>
            <given-names>G.</given-names>
            <surname>Amato</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Falchi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gennaro</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Rabitti</surname>
          </string-name>
          ,
          <fpage>Yfcc100m</fpage>
          -hnfc6:
          <article-title>A large-scale deep features benchmark for similarity search</article-title>
          , in: L.
          <string-name>
            <surname>Amsaleg</surname>
            ,
            <given-names>M. E.</given-names>
          </string-name>
          <string-name>
            <surname>Houle</surname>
          </string-name>
          , E. Schubert (Eds.),
          <source>Similarity Search and Applications</source>
          , Springer International Publishing, Cham,
          <year>2016</year>
          , pp.
          <fpage>196</fpage>
          -
          <lpage>209</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref4">
        <mixed-citation>
          [4]
          <string-name>
            <given-names>H.</given-names>
            <surname>Larochelle</surname>
          </string-name>
          ,
          <string-name>
            <given-names>Y.</given-names>
            <surname>Bengio</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Louradour</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Lamblin</surname>
          </string-name>
          ,
          <article-title>Exploring strategies for training deep neural networks</article-title>
          .,
          <source>Journal of machine learning research 10</source>
          (
          <year>2009</year>
          ).
        </mixed-citation>
      </ref>
      <ref id="ref5">
        <mixed-citation>
          [5]
          <string-name>
            <given-names>Y.</given-names>
            <surname>Bengio</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Lamblin</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Popovici</surname>
          </string-name>
          ,
          <string-name>
            <given-names>H.</given-names>
            <surname>Larochelle</surname>
          </string-name>
          ,
          <article-title>Greedy layer-wise training of deep networks</article-title>
          ,
          <source>in: Advances in neural information processing systems</source>
          ,
          <year>2007</year>
          , pp.
          <fpage>153</fpage>
          -
          <lpage>160</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref6">
        <mixed-citation>
          [6]
          <string-name>
            <given-names>D. P.</given-names>
            <surname>Kingma</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Mohamed</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D. Jimenez</given-names>
            <surname>Rezende</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            <surname>Welling</surname>
          </string-name>
          ,
          <article-title>Semi-supervised learning with deep generative models</article-title>
          ,
          <source>Advances in neural information processing systems</source>
          <volume>27</volume>
          (
          <year>2014</year>
          )
          <fpage>3581</fpage>
          -
          <lpage>3589</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref7">
        <mixed-citation>
          [7]
          <string-name>
            <given-names>Y.</given-names>
            <surname>Zhang</surname>
          </string-name>
          ,
          <string-name>
            <given-names>K.</given-names>
            <surname>Lee</surname>
          </string-name>
          ,
          <string-name>
            <given-names>H.</given-names>
            <surname>Lee</surname>
          </string-name>
          ,
          <article-title>Augmenting supervised neural networks with unsupervised objectives for large-scale image classification</article-title>
          ,
          <source>in: International conference on machine learning</source>
          ,
          <year>2016</year>
          , pp.
          <fpage>612</fpage>
          -
          <lpage>621</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref8">
        <mixed-citation>
          [8]
          <string-name>
            <given-names>G.</given-names>
            <surname>Lagani</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Falchi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gennaro</surname>
          </string-name>
          , G. Amato,
          <article-title>Hebbian semi-supervised learning in a sample eficiency setting</article-title>
          ,
          <source>Neural Networks</source>
          <volume>143</volume>
          (
          <year>2021</year>
          )
          <fpage>719</fpage>
          -
          <lpage>731</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref9">
        <mixed-citation>
          [9]
          <string-name>
            <given-names>G.</given-names>
            <surname>Lagani</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Falchi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gennaro</surname>
          </string-name>
          , G. Amato,
          <article-title>Evaluating hebbian learning in a semisupervised setting</article-title>
          ,
          <source>in: International Conference on Machine Learning, Optimization, and Data Science</source>
          , Springer,
          <year>2021</year>
          , pp.
          <fpage>365</fpage>
          -
          <lpage>379</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref10">
        <mixed-citation>
          [10]
          <string-name>
            <given-names>G.</given-names>
            <surname>Lagani</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Bacciu</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gallicchio</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Falchi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gennaro</surname>
          </string-name>
          , G. Amato,
          <article-title>Deep features for cbir with scarce data using hebbian learning</article-title>
          ,
          <source>arXiv preprint arXiv:2205.08935</source>
          (
          <year>2022</year>
          ).
        </mixed-citation>
      </ref>
      <ref id="ref11">
        <mixed-citation>
          [11]
          <string-name>
            <given-names>D. P.</given-names>
            <surname>Kingma</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Ba</surname>
          </string-name>
          ,
          <article-title>Adam: A method for stochastic optimization</article-title>
          ,
          <source>arXiv preprint arXiv:1412.6980</source>
          (
          <year>2014</year>
          ).
        </mixed-citation>
      </ref>
      <ref id="ref12">
        <mixed-citation>
          [12]
          <string-name>
            <given-names>G.</given-names>
            <surname>Lagani</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Falchi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Gennaro</surname>
          </string-name>
          ,
          <string-name>
            <surname>G.</surname>
          </string-name>
          <article-title>Amato, Comparing the performance of hebbian against backpropagation learning using convolutional neural networks</article-title>
          ,
          <source>Neural Computing and Applications</source>
          <volume>34</volume>
          (
          <year>2022</year>
          )
          <fpage>6503</fpage>
          -
          <lpage>6519</lpage>
          .
        </mixed-citation>
      </ref>
    </ref-list>
  </back>
</article>