Unsupervised 3D Object Learning through Neuron Activity aware Plasticity
Authors: Beomseok Kang
Biswadeep Chakraborty
Saibal Mukhopadhyay
Venue: ICLR 2023 poster (2023)
Keywords: Hebbian learning

Abstract: We present an unsupervised deep learning model for 3D object classification. Conventional Hebbian learning, a well-known unsupervised model, suffers from loss of local features leading to reduced performance for tasks with complex geometric objects. We present a deep network with a novel Neuron Activity Aware (NeAW) Hebbian learning rule that dynamically switches the neurons to be governed by Hebbian learning or anti-Hebbian learning, depending on its activity. We analytically show that NeAW Hebbian learning relieves the bias in neuron activity, allowing more neurons to attend to the representation of the 3D objects. Empirical results show that the NeAW Hebbian learning outperforms other variants of Hebbian learning and shows higher accuracy over fully supervised models when training data is limited.

Published as a conference paper at ICLR 2023
UNSUPERVISED 3D OBJECT LEARNING THROUGH NEURON ACTIVITY AWARE PLASTICITY
Beomseok Kang, Biswadeep Chakraborty & Saibal Mukhopadhyay School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA 30332, USA (beomseok, bi biswedepp, smukhopadhyay wawpaaacyyygttt..eeace...e
ABSTRACT
We present unsupervised deep learning model for 3D object classification. Conventional Hebbian learning, a well-known unsupervised model, suffers from loss of local features leading reduced performance for tasks with complex ge- ometric objects. We present a deep network with a novel Neuron Activity Aware (NeAW) Hebbian learning rule that dynamically switches the neurons to be gov- erned by Hebbian learning or anti- Hebbian learning, depending on its activity. We analytically show that NeAW Hebbian learning relieves the bias in neuron activity, allowing more neurons to attend to the representation of the 3D objects. Empir- ical results show that the NeAW Hebbian learning outperforms other variants of Hebbian learning and shows higher accuracy over fully supervised models when training data is limited.
INTRODUCTION
Supervised deep networks for recognizing objects from 3D point clouds have demonstrated high accuracy but generally suffer from poor performance when labeled training data is limited (Wul al., 2015 Qi et al. 2017a Wang et al.) 2019 Maturana & Scherer 2015). On the other hand, self-supervised or unsupervised models can be trained without labeled data hence improving the performance in data efficient scenarios. Sele-super learning methods have been studied for 3D object recognition mostly in an autoencoder setting, which necessarily reconstructs input to learn the representation (Achlioptas et al. 2018, Girdhar et al., 2016). Unsupervised learning has also been applied to pre-process the input for an encoder but still largely relying on supervised learning (Li ct al., 2018). Conventionally, self-organizing maps and growing neural gas have been used as fully unsupervised learning for 3D objects while they aim to reconstruct the surface of the objects (do Régo et al.) 2007: Mole & Araújo 2010). A fully unsupervised deep network for 3D object classification has rarely been studied.
Unsupervised Hebbian learning is known to offer attractive advantages such as data efficiency, noise robustness, and adaptability for various applications (Najarro & Risi 2020 Kang al., 2022 Mi- coni et al.) 2018)] Zhou et al., 2022). The basic Hebbian and anti- learning refer to that synaptic weight is strengthened and weakened, respectively, when pre- and post-synaptic neurons are simultaneously activated (Hebb) 2005). Many past efforts have developed variants of Hebb's 's rule. Examples include Oja's 's rule and Grossberg's 's rule OOja 1982, Grossberg 1976) for object recog- nition (Amato et al. 2019 Miconi 2021). ABCD rule (Soltoggio et al. 2007) for meta-learning and reinforcement tasks (Najarro & Risi 2020), and another variant for hetero- aastoiiissoeiitivee memory [Limbacher & Legenstein (020). However, Hebbian learning is often vulnerable to the loss of local features (Miconi [022] Bahroon et al. 2017: Bahroun & Soltoggio 2017 Amato et al., (019). This is a major challenge for applying Hebbian rules for tasks with more complex geometric objects, such as object recognition from 3D point clouds.
In this paper, present an unsupervised deep learning model for 3D object recognition that uses a novel neuron activity-aware plasticity-basee Hebbian learning to mitigate the vanishing of local features, thereby improving the performance of 3D object classification. We observe that, in net- works trained with plain Hebbian learning, only a few neurons always activate irrespective of the object class. In other words, spatial features of 3D objects are represented by the activation of only

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few specific neurons, which degrades task performance. We develop hybrid Hebbian learning rule, referred to as the Neuron Activity Aware (NeAW) Hebbian, that relieves the biased activity. The key concept of NeAW Hebbian is to dynamically convert the learning rule of synapses associated with an output neuron from Hebbian to anti-Hebbian or vice versa depending on the activity of the output neuron. The reduction of bias allows different subset of neurons activate for different object classes, which increases class-to-class dissimilarity in the latent space. Our deep learning model uses feature extraction module trained by NeAW Hebbian learning, and a classifier module is trained by supervised learning. The feature extractor designed as multi-layer perceptron (MLP) transforms the positional vector of sampled points on 3D objects into high-dimensional space (Qi et al. b). The experimental results evaluated on ModelNet10 and ModelNet40 (Wu et al., 2015 show that the proposed NeAW Hebbian learning outperforms the prior Hebbian rules for efficient unsupervised 3D deep learning tasks. This paper makes the following key contributions:
We present deep learning model for 3D object recognition with the NeAW Hebbian learn- ing rule that dynamically controls Hebbian and anti-Hebbian learning to relax the biased activity of neurons. The NeAW Hebbian learning efficiently transforms spatial features of various classes of 3D objects into a high-dimensional space defined by neuron activities.
We analytically prove that the NeAW Hebbian learning relieves the biased activity of output neurons if the input is under a given geometric condition, while solely applying Hebbian or mebi-Hebbian learning does not guarantee the relaxation of the skewed activity.
We analytically prove that purely Hebbian learning and anti-Hebbian learning on the biased neuron activity leads to a poor subspace representation with few principal components, thereby limiting the performance in the classification tasks.
We empirically demonstrate that the NeAW Hebbian learning rule outperforms the exist- ing variants of Hebbian learning rules in the 3D object recognition task. We also show that NeAW Hebbian learning achieves higher accuracy than end-to-end supervised learn- ing when training data is limited (data-efficient learning).
2 RELATED WORK AND BACKGROUND
Deep Learning Models for 3D Object Recognition Supervised 3D convolutional neural network (CNN) models have been developed to process a volumetric representation of 3D objects (Maturana & Scherer, 2015 Wu et al. 2015) but high sparsity in volumetric input increases computation cost, low-resolution point clouds. Multi-view CNN renders 3D objects into im- ages at different views and processes these images using 2D CNN (Su et al. 2015) VGG and ResNet-based models show good performance when the training images are well-engineered under proper multi-views (Su et al., 2018). However, 2D CNN-based approaches are difficult to scale to complex 3D tasks (Qi al. (017a). Recently low-complexity point-based models have been designed to process a point cloud where input is (x,, y, z) coordinates of points (Qi et al. 7a). Self-supervised learning models have also been proposed based on autoencoder and generative ad- versarial networks (Wu et al. 2016; Sharma al. 2016). Both models accept a voxel representation of 3D objects and learn the latent representation of the objects by reconstructing the voxel. The learned representation is used as the input to additional classifier. Our unsupervised learning does not use labels, but unlike existing self-supervised models, our approach does not reconstruct the objects.
limiting applications
Hebbian Learning Models The variants of the Hebbian learning rule are given as:
w(t) + nyx
Hebb's rule
w(t + 1)
w(t) + ny(x vw(()) Oja's rule
(I)
w(t) + my(x w(t) Grossberg's rule
where x, y. and are the input, output, and weight, 1) is the learning rate. Hebb's rule is the basic form of Hebbian learning where weights are updated if both the input and output neuron fire (Hebb. 2005). This linear association component analysis (PCA) if the data samples are assumed to be zero-mert mean (Weingessel & Homnk 2000). However, the plain Hebb's 's rule is often vulnerable the divergence of weight vectors as
be interpreted as biologically plausible principal
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Lateral
[3, 64, 128, 1024)
Inhilittio
MLP layers
(1024, 512, 256, k) MLP
Number of points
30C e
Inactivated neuron
"Chair" scores)
Shared
Gross paint inhibition
Activated neuran
Point cloud with points
O
Figure 1: 3D object classification model.
there is explicit upper bound (Porr& & Wörgötter 2007 Oja's j rule has an additional constraint term on Hebb's rule to normalize the weight vectors Ojja 1822). It derived by dividing the weight update in plain Hebb's rule by its norm, assuming that the input and output are linearly associated. Grossberg's 's rule is another variant where a constraint term enables the weight to converged on the input; hence the weight is more aligned with the frequently observed data (Grossberg 1976). Recent works have used Grossherg's ' rule for image feature extraction (Miconi 2021 Amato et al. 2019),
We can also modify Grossherg's 's rule be identical to Kohonen's 's self self-organizing map (SOM) by applying Winner-Take. three-dimensional spatial features (Kohonen) 2012 Li rule with the notion of neuron activity is recently studied for learning 2D point sets (Kang et al., 2022]. In this paper, we mainly compare NeAW learning with Hebb's s rule, Grossberg's rule, and Oja's rule and demonstrate the proposed rule is superior to them in the 3D object classification task.
(WTA) to output neurons and can be used for learning two-dimensional or
2018). Another variant of Grossberg's
PROPOSED APPROACH
3.1
3D OBJECT CLASSIFICATION MODEL.
We process (a., 2) coordinates of points in point cloud by shared three-layer MLP with com- petition layers. NeAW learning is applied on the MLP (i.e. encoder), and the other layers (i.e. classifier) are trained by supervised learning. The output vector for a point is written by:
x4++1 (TA
where xt,, W and 12 are the input, weight, and output vector at layer 4, respectively. Winner- lake-all (WTA) is applied on the output of each layer so that the single output neuron with the highest similarity (i.e. the lowest |x W for x Aet1) has non-zero value while others forced to be zero (Coates et al. 2011 Hu et al. 2014; Miconi 2021). Figure shows the architecture of our model. There are types of inhibition for the last output neurons in the encoder. Each WTA layer makes lateral inhibition, as shown in the figure, and cross-point inhibition is applied to the last output neurons after the lateral inhibition. Then, the model aggregates the transformed vector by summing the elements of the remaining neurons across the columns. The cross- tross-inhibition and the summation can interpreted as a MaxPooling layer that directly finds the maximum value of each column. The proposed model consequently arrives at a PointNet-style model which combines the MLP and MaxPooling layer in the encoder design. The aggregated activation of the last neurons, the global feature of objects, is processed by FC layers with ReLU and Softmax activation in the classifier. LayerNorm layers are included between the FC layers. We observe the WTA modules play important role in the representation learning. The proposed learning in the model without the WTA modules is discussed in Appendix E.
3.2 MOTIVATIONAL. STUDY
The main motivation of the paper is to demonstrate the unsupervised learning rule that can balance the neuron activity and its theoretical analysis in the 3D object classification task. Figure shows
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0.16
0.24
0.16
NeAW-H learning
NeAW-aHt
NeAW
learning
Neuron Activity (pii)
learning
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Neuron Activity
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Index (7) (a)
(1)
Neuron (1)
(b)
(c)
Figure 2: The average neuron activity of each output neuron for ModelNet10 with (a) NeAW learn- ing with only positive weight update (Hebbian-like), (b) NeAW learning with only negative weight update Ganti-eebbian-like and (c) NeAW learning.
the average neuron activity in the last layer of the encoder with different learning rules. Note, the details of the figure are explained again Section 4. We measure how frequently each output neu- activates across different points, so-called neuron activity, to quantify whether the transformed vectors are distinguishable in high-dimensional space or merged to a simple global feature. We ob- serve that having only positive or negative weight update in the NeAW Hebbian learning lead few specific neurons in the last layer to frequently activate regardless of objects, Given input vectors are projected onto weight vectors, the biased activation of output neurons indicates the small number of principal weight vectors in high-dimensional space. In other words, multiple input points are associated with few weight vectors, which significantly decreases the variance of context vectors with different labels and consequently leads to classification failure. Hence, the skewed activity's relaxation is considered the key problem to solve this paper.
3.3 NEURON ACTIVITY AWARE HEBBIAN LEARNING
Definition 1. Neuron activity pli) of the j-th output neuron is the number of the activation divided by the number of entire points. The neuron activity is the neuron activates for all the points in point cloud, and 0 if the neuron never activates.
Definition 2. Activation boundary is the plane with the same Euclidean distance from two weight vectors that determine which neuron of the weight vector to activate.
Our learning rule is given by:
N-1
W W.y(t) f(pp" p6))
Σ
1 arg min x W.,(t))la (x; W.)(t)
(3)
N
1=
W. = -W) f(p" - p()) N
(xx W.)
(1 (xt W.)(())
TT"">p'' (t) 6 W.,(t)) ifp < pp)))
(4)
where W ".((t) is the weight vector of the j-th output neuron at f, x; the i-th input vector in a point cloud, p" is the optimal neuron activity. As WTA layers in the proposed model search the closest weight vectors from the input vectors, the learning rule is motivated by Grossberg's 's rule and the prior work (Kang al. 2022). The optimal neuron activity is given by 1/d where d is the number of output neurons (Kang et al. (022). The indicator function is considered another WTA. Each weight vector searches the closest input vector with regard Euclidean distance and is moved closer or further to the input vector depending Hebbian and Meii-eebbian learning using the indicator function f. indicating Hebbian learning for low-activity neurons (p" > p(i) and anti- Heii-Hebiin learning for high-activity neurons (p" < p())). Note. the weight is not updated if the neuron activity optimal. The learning rule is simplified to {4) if the k-th input vector is closest to the weight vector. We introduce a constant a and in the equation to vary the importance of Hebbian and anti-Hebbian learning. In addition to the NeAW
the range of pl.) It describes the conversion between
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activation boundary ( W. (t DII, [x W. (t 1
NcAW-aH learning
learning
learning
W..
W.y
W
W
W
W.
W W.
W
[x W, 1) x
[| W. well < ||x tt
[|x: W_)( < ||x W.r
Ix W W,,, 1 >
[x] - W (t 1)l,
|x W tt (d)
(a)
(b)
(c)
Figure 3: Geometry of input and weight vectors. (a) the initial geometry of the vectors and the changed geometry of the vectors after (b) NeAW-H learning, (c) NeAW-aH learning, and (d) NeAW learning.
Hebbian learning rule, define NeAW-H learning (a=1 and b=-1) and NeAW-aH learning (a=-1 and b=1) by making the learning rule only has either positive or negative (i.e. non-hybrid) weight update regardless of the activity. Note, NeAW Hebbian learning does not refer NeAW-H learning. We observe that balancing the neuron activity is also an important function in the brain (Keck et al.) 2017 Turrigiano & Nelson, 2004). The biological plausibility of the proposed learning rule is discussed in Appendix G.
An important question is how the hybridization of the learning rules depending the neuron activity effectively relaxes the biased neuron activity. In a simplified scenario, we can consider two weight vectors and an input vector with the assumption that the corresponding neuron of each weight vector is highly active or non-active. Also, and b are assumed to be 1. The problem is whether NeAW Hebbian learning shifts the activation boundary so that the other neuron activated after learning.
To help the intuitive understanding of how NeAW Hebbian learning changes the activation boundary, the geometry of the three vectors described in Figure 3 In Figure aa), the activation boundary as a dotted line shows that input is left to the boundary plane, and the j-th neuron is assumed to be highly active while the jth neuron is not active before learning. We expect that the learning properly shifts the activation boundary to activate the j'-th neuron and relieve the biased activity. The geometry the vectors after NeAW-H, NeAW-aH, and NeAW Hebbian learning are compared in Figure b)), (c), and (d), respectively. The activity-agnosti rules move both the weight vectors to be closer to or further from the activation boundary, implying the activation boundary cannot be effectively shifted to either left right space. It is desirable that both the weight vectors also shifted the same direction. In this light, NeAW Hebbian learning offers the appropriate shift of the activation boundary and consequently switches the winner neuron from the highly active j-th neuron the non-active -th neuron, shown in Figure d).
Based on this idea, we develop formal theorems and corollaries to mathematically prove that NeAW Hebbian learning changes the winner neuron while the others cannot:
Theorem 1. Let two weight vectors W.. and W '.g" with the biased activity and an input vector X satisfy |X; (t) < ||x;
W
o (t) at t. Then, NeAW Hebbian learning changes the sign
1)ll at t. + if the vectors are on the of the inequality to ||x; W., 1) > Xx; W ". 4x W W. (1)l2.
geometric condition (1 ]] W.
Proof Sketch. (Formal proof in Appendix A) Due to the biased activity, each weight vector is updated by Hebbian learning or anti-Hebbian learning. Then, we have update rules, W. W (t 1)
W., (f)) for the low-active j'thh neuron. From the assumption, we have ||x; [ (t) #
W W..;(t) & (x, (t)) for the high-active j-th neuron and W W (t+1) = W
(t)l2 <
||x; W., (f) which indicates that the winner neuron is the j-th neuron at t. Using the two updates rule to substitute the weight vectors from to + 1, the inequality at t 1 is derived.
The sign in the inequality is flipped if the Euclidean distances satisfy (1 }){ (t)(t))l2
(1 |) Way (1)l22
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Corollary 1.1. Let two weight vectors W. so and W W' with the biased activity and input vector X; satisfy ||x; Wa:(t)]l2 < ||x; W a a (t)ll2 at t. Then, Hebbian learning (NeAW-H) preserves the sign of the inequality |x: W (t + 1)lla <
(t (t+ 1)ll2 at t
Corollary 1.2. Let two weight vectors Was and W.."" with the biased activity and an input vector
X; satisfy ||x; - Wll(t))l2 < l|x - W W. (t)ll2 at t. Then, anti- anti-Hebhian learning (NeAW-aH) preserves the sign of the inequality to Xx W.s(1 + 1)l12 < ||x:
- W. (t+ 1)112 al L 1+1.
Proof for corollaries is in Appendix A.
The prior study has shown that Grossberg's 's rule combined with WTT does k-means clustering (Hu et al.] (014)) While NeAW Hebbian learning does not aim to minimize the distance between the in- put and weight required for the clustering, we can interpret the activated neurons cluster centroids. The more output neurons the model activates with the balanced activity, the smaller spatial features are responsible for each neuron if they create separable clusters. In other words, more activated neurons are preferred to avoid the vanishing of local features, which can be achieved by maximizing the variance of the neuron activity. The variance at layer 4 is written by:
N
1
Varly")
(y
-yyh)
(yy Hyyt))
N
k=0
(5)
N-1
N N2
N
Σ
y
yk
y
1
N
N2
k,
R,
where y/ is a binary vector whose element is either 0 or 1. It obtained from (2) by applying a step function on it only to consider whether the neuron is activated or not. Also, y y is always 1 as the binary vector has single non-zero element due to WTA. The proposed NeAW Hebbian learning eventually aims to increase the variance of the activated neurons so that each neuron is responsible for the different geometric parts of 3D objects by properly rotating the weight vectors through the neuron activity aware plasticity, or more formally:
Theorem 2. For a network of N neurons in the biased activity with weight vectors W and an inpur vector x, the variance of the distribution of activated neurons follows Var(NeAW Hebbian) > Var(Hebbian) ) or Var(anti - Hebbian) which means NeAW Hebbian learning provides the greatest variance of the neuron activity compared solely using Hebbian or anti-Hebbian learning.
Proof Sketch. (Formal proof in Appendix A) First, we note that Hebbian (H) and anti- Hebbian (aH) networks optimize the information capacity by orthonormalizing the principal subspace (Plumbley 1993j. To prove that the neuronal activations for the hybrid learning rule have a greater variance than the standard H and aH learning rules, first assume that a network with greater number of neuronal activations can encode a large number of principal components in the subspace. We consider that the local H and aH learning rules recover a principal subspace from the data using principled cost function (Pehlevan et al. 2015). Hence, we rewrite the problem as:
span (LNeAw) 2 span (Crr or span (Cat1)
(6)
where LH and Lalt denote, the subspace learned using the Hebbian and anti- anti-Hebbian learning rules, respectively. Let us consider the case where we have N neurons learned using H or aH learning rules. Without loss of generality, using the learning method L and the metric space of the neuron vectors and the weight vectors to be represented as W. For the n + 1-th neuron, the neuron can be trained using either the same learning rule L or the complementary learning rule L. Thus, we study the differential gain from using one learning rule over the other for the 1-th neuron. learning rule extracts the principal components of the input space by projecting the input space into the orthogonal subspace, and neural network with larger number of principal components can learn a better representation of the input space. Again, H and aH learning rules lead to different principal components (Rubner & Tavan 1989 Carlson 1990). Hence, we show that using NeAW Hebbian learning approach leads to increase in the number of the principal components learned, which leads greater span of the learned subspace for the hybrid learning model compared to the homogeneous H or aH learning method, which, in turn, entails higher variance for the neuronal activity.
can assume that the first (n N) neurons are trained

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Hebb's rule
Grossberg's
Oja's rule
rule
(8)
(b)
(4)
Figure 4: I-SNE for ModelNet1O in (a) Hebb's rule, (b) Grossberg's rule, (c) Oja" rule, and (d)
NeAW learning.
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Accuracy
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0.94
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ModelNet40
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(NeAW)
ModelNet10
0.86
ModelNet40
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20
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40
50
20
40
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9 50
Epochs (a)
Epochs (b)
Epochs
Epochs
(c)
(d)
Figure 5: Effect of training with regard to the variance of neuron activity and test accuracy. (a) the variance of neuron activity with the different learning rules in ModelNet10 and (b) ModelNet40 and (c) the test accuracy at different epochs for ModelNet and (d) ModelNet40.
EXPERIMENTAL RESULT
Experimental setting The proposed model is evaluated on ModelNet10 and ModelNet40 datasets, which include 10-class and 40-class 3D CAD objects for 3D deep learning (Wu et al.] 2015). We randomly sample 1024 points on the surface of the objects. The encoder trained by unsupervised learning with unlabeled datasets while the classifier is trained by supervised learning with object labels. Note, the weights of the encoder are not changed during training the classifier. For all the unsupervised learning rules, we set training epoch 50, learning rate le-2, and train batch 4; however, learning rate is proportionally increased when the amount of training data is limited. For example, learning rate is le-1 for 10% training data and 4e-2 for 25% training data. For the classifier, use training epoch 100, learning rate 1e-3, and train batch 32 for both ModelNet10 and ModelNet40. We report average instance accuracy for test datasets.
Balancing of Neuron Activity Figure 2 shows the neuron activity of each output neuron for the activity-agmostic and activity-aware learning in the same model. x-axis the figure is the index of output neurons, and y-axis is the normalized number of the neuron activation for ModelNet10 test dataset. The activity is first summed over the all test samples and then normalized by the entire activities. In NeAW-H and NeAW-aH learning cases, the activity is dominated by few neurons, implying only few neurons are used to represent the objects. On the other hand, NeAW learning clearly displays the more distributed activity than activity-agnosti cases. In addition. the activity level increases for non-active neurons and decreases for high-active neurons, enabling more output neurons contribute to the representation. We also study how the neuron activity differs depending on the object classes. Figure 1O and [ Appendix C show the average neuron activity for the 10 object classes in NeAW learning and NeAW-H and NeAW-aH learning. displays that the neurons in NeAW learning object-dependently activate. thereby, increasing the dissimilarity between the objects in the latent space. However, the neurons in the NeAW-H and NeAW-aH learning often activate regardless of the object classes. We study the correlation between the object-dependent activation and better representation learning in Appendix B. Also, the causal link between the biased activity and poor learning is studied in detail.

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Table 1: Comparison with other unsupervised learning rules.
ModelNet-10
ModelNet-40
Learning Rule
Variance
Accuracy (%)
Variance
Accuracy (%)
Hebb's rule
0.9663
72.22 + 0.46
0.9391
41.52 1.21
Grossberg's 's rule
0.8714
74.45 + 0.56
0.8693
55.96 + 0.39
Oja's rule
0.9182
80.40 + 0.03
0.8953
61.22 + 3.82
Ours NeAW-H 0.9165
74.19 + 0.21
0.9251
49.83 + 4.72
Ours NeAW-aH
0.8624
64.63 + 0.60
0.8511
35.65 + 0.80
Ours - NeAW 0.9793
88.22 + 0.62
0.9741
76.20 = 0.30
Evolution of Neuron Activity during Unsupervised Learning We study the evolution of neuron activity and test accuracy as a function of training epochs of the unsupervised encoder. In all the cases, the unsupervised encoder frozen after training with Hebbian learning for certain number of epochs, followed by full supervised training of the classifier. Figure 5 shows the variance of neuron activity during training of the model. We observe that variance quickly saturated to a high value in NeAW Hebbian learning. In comparison, the variance increases slowly and to lower value with NeAW-H learning and NeAW-aH learning, respectively. Figure (3c) and (d) represent the test accuracy for ModelNet10 and ModelNet40 different training epochs with NeAW Hebbian learning. We observe the variance and accuracy converge in 50 training epochs.
Comparison with other Hebbian learning Figure show t-SNE plots for the Hebb's rule, Gross- berg's rule, Oja's 's rule, and NeAW learning. It qualitatively indicates that the NeAW learning su- perior to the other Hebbian learning rules as the context vectors are well clustered with their labels. We also study the neuron activity distribution shown in Figure 2 for the other rules. The neuron ac- tivity of the NeAW learning is also well balanced than the other rules (see Figure Gi Appendix C.) The Grossberg's 's rule and Oja's rule still display the skewed neuron activity dominated by the few neurons while the Hebb's rule achieves the relatively balanced activity. However, we observe that the Hebb's rule has many neurons that activate regardless of the object, indicating the less object- dependent activation than the Grossberg's rule and Oja's 's rule as shown in Figure 12 in Appendix C. Table I compares the variance of the neuron activity and test accuracy with the plain Hebb's 's rule, Grossberg's 's rule, Oja's rule, and NeAW Hebbian learning. The test accuracy is evaluated trained models with 5 different random seeds, and the mean and variance of the accuracy are listed in the table. We observe the NeAW Hebbian learning shows the highest variance of the neuron activity and improved test accuracy in both the datasets. In particular, the accuracy gap increases in ModelNet40, implying the potential pitfalls of the other Hebbian learning rules on complex 3D object datasets. Note, though the plain Hebb's 's rule show the higher variance of the neuron activity, which indicates the balanced activity, than the Oja's rule, the test accuracy is lower. This is due to the unbounded norm of the weight vectors. In other words, instead of learning the points that distin- guish objects with different labels, commonly appeared points in the training samples continuously strengthens the certain weights (see Figure 20 in Appendix F.) It can increase the point-to-point variation but sacrifice local features.
the
Comparison with Existing Supervised Learning Models compares the proposed model with prior models for 3D object recognition. First, our model shows similar accuracy other self-supervised model (Wu et al. 2016 Sharma et al., 2016). Accuracy difference with 3D-GAN increases in ModelNet40 but 3D GAN needs reconstruct a (30, 30, 30) voxel to learn the represen- tation resulting in high computational cost for high-resolution 3D objects. Our model shows similar accuracy to other fully supervised voxel-based approaches while using 6.2x fewer parameters (Wu) et al. 2013). The accuracy difference with other large boset-based approaches is more pronounced in ModelNet40 but they use 1.8x to 4.3x more parameters than our model. We also compare un- supervised version and fully supervised version (i.e., the encoder is trained with backpropagation instead of Hebbian learning) of our model. The proposed unsupervised version shows a marginally lower accuracy from the supervised version. In addition, we observe that the unsupervised learning preserves the accuracy well in smaller models than the supervised learning. Figure [6 in Appendix E shows that 0.09MB unsupervised model achieves 72.5% for NedelNet40 while the supervised
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Table 2: Overall accuracy (%) and model size comparison with supervised models.
Model
Learning
ModelNet10
ModelNet40
Size (MB)
3DShapeNet Wu et al.
2015
Supervised
83.5
77.0
19.2
PointNet QQi et 2017a
Supervised
89.2
13.2
PointNet++ (Qi et al.
2017b
Supervised
91.9
5.6
DGCNN Wang et al.
2019)
Supervised
92.9
6.9
VConv-DAE (Sharma et al.
2016
Self-supervised
80.5
75.5
3D-GAN Wu et al. 2016)
Self-supervised
91.0
83.3
Ours Backpropagation
Self-supervised
92.2
79.5
4.1
Ours Backpropagation
Supervised Unsupervised
91.1
77.2
3.1
Ours NeAW Hebbian
88.2
76.2
3.1
NeAW Hebbian
NeAW Hebblan rule Grossberg's rule Hebb's rule
rule Grossberg's s rule
Backpropagation NeAW Hebblan
Backpropagation
Hebb's rule
NeAW Hebbian
61.1
90.3
84.9
ModelNet10
100
MadelNNt40
Accuracy (%)
80
88.483.4
81.3
79.4
80
Accuracy (%)
Br...
80
70
77.4
Accuracy
70
7036.2
61.5
70
60
71.0
60
60
64.6
60
50
50
50.5
42.9
50
ModelNett0
45.0
ModelNet40
35.5
30
100
50
25
10
100
50
25 10
100
50
25
5
1
100
50
10 5
Amount
(%)
Amount labeled data (%)
labeled (%)
Amount
(%)
(a)
(b)
(c)
(d)
Figure 6: Comparison with other learning rules in limited training data. (a) the accuracy of end- to-end supervised learning (backpropagation) and NeAW Hebbian learning for ModelNet10 and (c) ModelNet40, (b) and the accuracy of other unsupervised learning for ModelNet10 and (d) Model- Net40.
learning in the same model shows 59.9%. Our model also includes self-supervised learning while having more parameters and computational costs (Appendix E).
Learning with Less Training Data. We study the performance of NeAW Hebbian as the amount of the training samples decreases (Figure 6). We randomly drop samples from each class based on the percentage amount the labeled data. For the other Hebbian learning rules, NeAW Hebbian learning always shows higher accuracy in both ModelNet10 and ModelNet40. As shown in Table NeAW Hebbian learning achieves marginally lower accuracy to fully supervised model (denoted as Backpropagation) when the all data is used for training. However, in all other cases with less than 25% labeled data, NeAW Hebbian learning shows higher performance than Backpropagation. Our model also shows better performance than alternative Hebbian rules at reduced training data.
5 CONCLUSION
We present the neuron activity aware (NeAW) plasticity in Hebbian learning that combines Hebbian and anti-Hebbian learning to relieve the biased neuron activity. Multi theorems and corollaries are developed to understand how NeAW Hebbian learning relieves the biased neuron activity and the correlation between the balanced activity and better representation learning in the classification task. Experimental results demonstrate that NeAW Hebbian learning achieves higher accuracy than other Hebbian learning rules and supervised learning with limited training data. We believe the paper provides a theoretical understanding of how neuron activity can be used for modulating the learning rule and its application to unsupervised 3D learning.

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ACKNOWLEDGMENTS
This material is based on work sponsored by the Office of Naval Research under Grant Number N00014-200 104-00-1-2444 1-2422. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed implied, of the Office of Naval Research or the U.S. Government.
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A PROOF OF THEOREM
Theorem 1. Let two weight vectors W W.j and W 1. with the biased activity and an input vector X satisfy |x; - W.,(tt)l2 < ||x; W. o (f) at t. Then, NeAW Hebbian learning changes the sign
1)ll at t +++ if the vectors are on the geometric condition (1 + }) W. (2 (1 ) W -
of the inequality to ||x; W. + 1) > |x W ".
+
Proof. Let's say there are an input vector X,, corresponding winner is the j-th output neuron (i.e. j arg min||x; (t)(,))l,, and the other loser is the j -th output neuron. We are interested in how the hybrid update changes the activation boundary. Our learning rule is given by:
W = W (1)(() s(p" -D0)) N
Σ 1. e arg min x (0ll W.)(!)
Given there is single input vector, the learning rule can be simplified by:
- - 17 (x: (t)(t))
W As we are interested in the hybrid update relieves the biased activity such as too high activity or zero activity, the j-th and j -th neuron are assumed have higher pوu) and lower p than p". Then, with an assumption that and b are to simplify the case, the equation is again written as:
(8)
W.y(t+ (
7 1(x Wt)(t))
N
W 1) NO++NN
(x. (t)((t)
(10)
Our goal is to show that the learning rules change the activation boundary. At before learning, the Euclidean distance for the j-th neuron lower than the j' -th neuron, or more formally:
|X (t) < ||x W. (t)ll
(11)
xx Wt)(t)11 ||x: W (t)l <0
(12)
(t + W W (2)-) (W. (t)
W .''' (f)) < 0
(13)
At t + after learning, the left-hand side 13 is written as:
( +1) + W ' (1+ 2x.) (W (1
W W
(1+ 1))
(14)
The hybrid update changes the winner neuron if it is positive. Using re-written as:
and (10
(4) can be
(W (t) + W W (t) 2x, + ( (W. W (t)}
7
N
(15)
(W (W.y(t) - W. (t) N
(2x; Wty(t) W W (t))
41 N
|4 W (t) + +(1
7 ) (t) + W w. -2x,) (t) W (t)
(16)
N
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Substituting the second term in (16) by the left-hat reft-hand side of
(1 + N Plx: W.y (1 N |1x: W (0)
(17)
Equation 14 is positive if
X W WO
(1+ )²
(18)
W N
Corollary 1.1 Let two weight vectors W oo and W W.g with the biased activity and input vector
x; satisfy X
(.)ll)l2 ||x W o (l)l22 2 at I. Then, Hebbian learning (NeAW-H) preserves
the sign of the inequality |x W + 1)ll2 < - (t + 1)1l at
Proof. Similar to the proof of Theorem 1, we need to show the equation below is positive if Hebbian learning changes the winner neuron:
(W. , (1 +1 (1++1) - ((2,))
( W w. (1+ 1))
(19)
We have Hebbian learning rules for both the j-th and j th neuron as:
W.y(t = (t =
y
(x; W.y(t)
(20)
N
W (
N (x; w.)
(21)
Using the learning rules re-write 19
(W + W W (1) 2x; + (2x; W.y(() - W. o
N
(22)
(t) - W., (t) N
(W .. (t) W (t)}
(1 N (W W o (1) 2x,) (W.,(() - W.y
(23)
As assume
(() + W (1) (1) 2x;) (W. W ' (1) <
(24)
Equation
19 is always negative. Thus Hebbian learning cannot change the winner neuron.
Corollary 1.2
Let two weight vectors W., and W with the biased activity and an input vector (t) < |x, - W "a (t)ll2 at t. Then, anti- onti-Hebbian learning (NeAW-aH)
X; satisfy x
preserves the sign of the inequality to x W.a + 1)l12 < |x W + 1)ll2 at + 1.
Proof. Similar to the proof of Theorem we need show the equation below is positive anti- Hebbian learning changes the winner neuron:
(W J (t +1 W. W (t+ 1) - 2x4) (W + ( (t+ 1))
(25)
We have anti- anti-Hebbian learning rules for both the j-th and ý -th neuron as:
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W. - N/1
N N(x: W.y(()
(26)
W (.y((++)==W, 55 (t)
1 [x, W. w
(27)
Using the learning rules to re-wir re-write
25
(W))(1) + W 'f (t) 2x; + (2x; W.,(1) W (.)) N
(28)
(W.,(t) - W. (t)
2 (W - W (()}
"
N
(1- / (W.((()) + W off' (t) 2x,) (W.j(t) - W (t)
(29)
As assume
(1) + W '. (1) 2x,) (W.j(0) .f (1)) <0
(30)
Equation 25 is always negative. Thus anti-Hebbiie Hebbian learning cannot change the winner neuron.
Theorem 2. For a network of N neurons in the biased activity with weight vectors W and an in- put vector x, the variance of the distribution of activated neurons follows Var(NeAW Hebbian) > Var(Hebbian)) or Var anti Hebbian) which indicates NeAW Hebbian learning provides the great- est variance of the neuron activity compared to solely using Hebbian or anti aati-Hebbian learning.
Proof. First, note the results shown by Plumbley 1993), that SatiiiannaatiHHebbbbi networks optimize the information capacity by orthonormalizing the principal subspace. In order to prove that the neuronal activations for the hybrid learning rule have greater variance than the normal Hebbian or anti-Hebbian learning rules, we first assume that a network with a greater number of neuronal activations can encode a greater number of principal components in the subspace. We consider that the local Hebbian and anti-Hebbian learning rules recover a principal subspace from the data using a principled cost function shown by Pehlevan al. (2015).
The subspace method of classification is pattern recognition method where the primary model for class is linear subspace of the Euclidean pattern space (Watanabe & Pakvasa (973). Let u; R" = 1.. m < n be set of m linearly independent vectors that spans the subspace L as:
P P (w ..., Um) (
aqui
where Or; is some scalar. That is, essentially, the problem can be re-written as:
Var(NeAW Hebbian)
2 Var(Hebbian)) or Var Hebbian) 2 span (CHebblan)))) or span -Satt-Hebbiin)
span (CNeAW Hebbian)
(31)
where span is the minimum spanning subspace, i.e., the subspace does not have smaller sub- space spanned by a subset of the samples. For detailed explanation of minimal sample subspace refer to the works of Zhang & Xia (2019).
As shown before, the learning rules for the Hebbian and anti-Hebbian cases are given, respectively as:
Willebb (t+ Willeb (t) + 77
(x; W.y(t))
N
Wemb-tHebb (1+ 1) Weani-tebb (1)
77
(xi Way(()
(32)
N
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Now, let us consider the case where we have N neurons which are learned using Hebbian anti- Hebbian learning rules. We aim to bifureate the sets of trained neurons into Hebbian and anti- Hebbian classes. Without loss of generality, we assume that the first (n NN neurons trained using the a learning method & E {LH = Hebbian, Latt = anti-Hebbian, ). Mathematically. do binary partition of the metric space of the neurons and the weights. If consider the metric space of the neuron vectors and the weight vectors to be represented as W, we define the partition from X + Y, Z where X denotes the space of all the vectors, Y denotes the space of vectors where all the neurons have already learnt (which is interpreted as coloring of the vector) using learning algorithm L. denotes the space of vectors where none of the neurons have been assigned learning rule, i.e., they can learn using either Li or Laft. We denote the span of the subspace learned using the learning method
denoted as span( span((ff.......??
Now, let us consider the + 1-th neuron. There might be two cases that might arise: (i) the neuron is trained using the same learning rule L with which the previous n neurons are trained. (ii) the neuron is trained using the complementary learning rule L. Thus, we study the differential gain by using one learning rule over the other for the n + 1-th neuron.
We know that a learning rule basically extracts the principal components of the input space by projecting the input space into the orthogonal subspace. Also, we know that a neural network with richer subspace, i.e., subspace with a larger number of orthogonal principal components, can learn a better representation of the input space than a subspace with a fewer number of orthogonal principal components.
Again, let us consider the subspaces learned by the Hebbian L and the anti-Hebbian Laf learning rules. From previous works (Rubner & Tavan 1989 Carlson 1990; SSchaaudolph & Sejnowski 1991) we know that anti-Hebbian and Hebbian learning rules lead to different principal components. As described by Schraudolph & Sejnowski [199]] while the built-in temporal smoothness constraint enables Hebbian neurons to learn the invariance classes, anti-Hebbian synapses have been used for lateral decorrelation of feature detectors and removal of temporal variations from the input. A set of Hebbian feature detectors whose weight vectors span the hyperplane would characterize the associated class of stimuli. The anti-Hebbian learning algorithm, however, provides a more efficient representation when the dimensionality of the hyperplane is more than half of the input space. This because, in that case, we require fewer normal vectors than spanning vectors for unique characterization. Since anti-Hebbian neurons remove the variance within stimulus class, they present a different output representation to subsequent layers than the Hebbian neurons. Thus, since the linear subspaces learned by the two learning rules are distinct, we can say that:
span(z ...., vn U x S span( ...... f
+ span (rannHebbian) or span (antinnnteeeea Heabbine S span (NeAW Hebbian)
(33)
Thus, we see that using hybrid Hebbian learning rule increases the span of the subspace learned, i.e, increases the variance of the neuron activations.
Corollary 2.1. Hebbian anti- Heibiin learning on the biased neuron activity leads to poor subspace representation than NeAW Hebbian learning
Proof. As we showed before in Theorem 2 in (3), we see that the span of the subspace learned by the NeAW Hebbian learning algorithm is greater than the span of the homogeneous Hebbian or anti-Hebbian learning method. Let us label each neuron as 'H' or 'aH' for Hebbian or anti- Hebbian learning methods depending on the learning algorithm it selects. Now, let us denote the distribution of the nodes marked 'H' and the distribution of the nodes marked 'aH' in the vector space of the neuron weights. We notice that since the total number of neurons is fixed, the total probability mass of these two distributions is also fixed. Thus, if there probability mass of the aH'' neurons is lower i.e., for the set of neurons n, E N, TP nz] > P|E n.]. Now, considering the full set of principal components that the entire network can learn using an infinite number of neurons. As keep decreasing the number of neurons, keep learning a subset of this ideal set of principal components. Since the sum of probability masses of the Hebbian and anti- Hebi-ae learning algorithms are fixed, if the distribution is heavily biased
lot more 'H' neurons, the
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towards just Hebbian learning, then the principal axes that could be learned from the Meti-Hebbian learning are not present. Thus, we represent the principal axes of the learning rule L as ag, then min (Σ ax) S min ( aNeAw), where X represents either Hebbian unti-Hebbian learning neurons. Hence, a biased learning algorithm leads to a poorer selection the principal axes, which leads to worse performance.
B CAUSAL RELATION BETWEEN NEURON ACTIVITY AND REPRESENTATION LEARNING
night stand
night stand
monitor
dresser
bethtubb
monitor
dresser
batthtub
toilet
table
bed
desk
chair
tallet
table
bed
desk chair
monitor
0.020
monitor
0.015
|P|; |IP liz
0.010
0.005
night
night
bed
-0.005
bethtub
bathtub
chair
0.015
chair
sofa
-.020
sofa
0.05
0.10
0.15
0.20
Variance
(a)
(b)
Figure 7: (a) Dissimilarity matrix for ModelNet10 after NeAW learning, (b) 1.2 norm change of the dissimilarity matrix by deactivating a neuron in NeAW leanring, and (c) dissimilarity matrix after learning using Hebb's rule. The x-axis in (b) indicates the variance of the deactivated neuron's activity for different object classes, and the y-axis is the L.2 norm change of the dissimilarity matrix by the deactivation.
Causal Relation between the Biased Activity and Poor Learning Hebbian- unsupervised learning rules with Winner-Take-All (WTA) be interpreted on the factors that affect the quality of clustering to build a causal connection between the neuron activity and the performance. It is well-known that (1) the number of cluster centroids and (2) their scparability play significant role in determining the quality of clustering (Kodinariya & Makwana 2013). If there are too few cluster centroids, irrelevant points will associated with the same cluster centroid. Likewise, if the cluster centroids are not separable, i.e., multiple centroids are positioned at the same location, the effective number of clusters will still low. the end, we need the enough number of well separable cluster centroids that behave as principal components for different local features in the object. In Corollary 2.1 (Appendix A), we have proved that Hebbian and nati-Hebbian learning leads biased neuron activity and hence, fewer principal components than NeAW learning. Fewer principal components indicate fewer separable cluster centroids which leads poor performance. Hence, using Corollary 2.1
clustering module. We focus
state that a biased activity in the Hebbian and anti-Hebbian learning causes poor performance. That is, the biased activity problem should be resolved to improve the performance. However, does necessarily indicate that the balanced activity leads to better performance.
Correlation between the Balanced Activity and Better Clustering The causal link between bi- ased activity and poor performance suggests that reducing activity bias may improve performance. However, it does not necessarily indicate that the only balancing activity causes a better perfor- mance. In particular, we observe that the "balanced activity with object-dependent activation" is correlated with the clustering quality. The idea behind is that the neurons that activate regardless of the object classes (no object dependency) will not capture the features to design the classification boundary. In this light, we design an additional experiment to quantify the change of the clustering quality depending how dependent neurons using the dissimilarity matrix. The dissimilarity matrix for ModelNet10 is given by 10x10 matrix, where each element the dissimilarity between the encoded vectors of the corresponding object
different object classes. We quantify the quality
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classes (see Figure a)).) We choose cosine similarity to calculate the dissimilarity. Let a and gi for the encoded vector for A and B objects. Then, the dissimilarity in the matrix (D) is given by D[A[[B] 1 _ [ We calculate the change of the L2 norm of the dissimilarity matrix by deactivating neurons one by one. The experiment results demonstrate that deactivating the neuron that abject-dependentee activates reduces the dissimilarity between the objects, indicating poor clustering quality. In contrast, deactivating the neuron that activates regardless of the object labels increases the dissimilarity, leading to better clustering quality (see Figure (b).) Note, the number of activating neurons during the experiment are same as we deactivate a single neuron at a time.
Figure (c) shows that the dissimilarity matrix in Hebb's rule, and all the matrix elements in NeAW are higher than Hebb's rule. It indicates that the object classes are easily distinguishable in NeAW. As many neurons activate regardless of the objects in Hebb's rule, the inner product between the en- coded vectors increases, and the cosine-based dissimilarity decreases leading to the poor clustering quality. That is, the encoded vectors with less common activation will have the higher dissimilarity, which indicates the better clustering quality. In the end, the higher dissimilarity is preferred; and we consider the 1.2 norm of the dissimilarity matrix to compare the cluster quality between the learning rules. The 1.2 norm for Hebb's rule, Grossherg's rule, Oja's rule, and NeAW learning are 0.630, 1.049, 1.313, and 1.482, respectively. Note, the accuracy of them for ModelNet10 are 72.22%, 74.45%, 80,40%, and 88.22%, respectively.
In summary, the biased activity causes a poor learning in the proposed model while the balanced activity is not a cause of the better performance. The additional experiment demonstrates that the clustering quality varies depending on both the activity balance and the neuron's object dependency.
C NEURON ACTIVITY DISTRIBUTION IN OTHER LEARNING RULE
0.040
0.040
Supervised learning (before training)
Supervised learning
Neuron Activity (pll)
Neuron Activity (pll)
(after training)
0.024
0.024
0.016
0.016
0.008
0
200
400
600
800
1000
200
400
600
800
1000
Neuron Index (a)
Neuron Index (f) (b)
Figure& 8: The average neuron activity of each output neuron for ModelNet10 (a) before and (b) after end-to-end supervised learning.
0.16
0.16
0.16
Hebb's rule
Grossberg's rule
Oja's rule
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Figure 9: The average neuron activity of each output neuron for ModeINet10 with (a) Hebb's rule, (b) Grossberg's rule, and (c) Oja's rule.
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We study the neuron activity distribution of the supervised model and the other Hebbian learning rules. The neuron activity here is the average number of the activation calculated by the same way in Figure 2
Neuron Activity in Supervised Learning Note, our end-to-end supervised model does not have the WTA modules as they are not differentiable while the other Hebbian learning rules are trained with the WTA modules. We observe that the neuron activity of the supervised model is evenly distributed before training as shown in Figure M(a). This will be related to the random initialization of the weights, which randomly activate the neuron through ReLU function. Figure (8b) displays that the activity is less balanced after training in the supervised model though there IS no highly skewed activity observed in the Hebbian and anti-Hebbian learning.
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Figure 10: Neuron activity for different object class in NeAW learning. 180 neurons are displayed among total 1024 output neurons as the others do not activate for all the test samples.
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Figure 11: Neuron activity for different object class in (a) NeAW-H learning and (b) NeAW-aH learning. 29 and 18 neurons for Hebbian and anti-Hebbian learning, respectively, are displayed among total 1024 output neurons as the others do not activate for all the test samples.
Neuron Activity in other Hebbian Learning We perform the experiment for the other Hebbian learning rules such as Hebb's rule, Grossberg's rule, and Oja's rule. Figure 9 shows the average neuron activity is also biased in Grossberg's rule and Oja's rule, which is similar with NeAW-H and Ne learning in Figure Hebb's 's rule has the relatively balanced activity than the others. However, it does not lead to the better accruacy summarized in Table We qualitatively observe that all the learning rules have the less balanced activity than the NeAW learning. It parallels with the variance of the neuron activity in Table T
Object-wiss Neuron Activity in Hebbian Learning Rules Figure 2 8 and 9 show the average neuron activity for all the test samples. However, the average neuron activity for the different object class is not presented. First, Figure and display the average neuron activity in NeAW learning and NeAW-H and Ne learning, respectively. Each column in the figure represents a neuron index, and row indicates object label. The value of elements is normalized, where the maximum
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Figure 12: The neuron activity for different object class in (a) Hebb's rule, (b) Grossberg's rule, and (c) Oja's rule.
value is (red) indicating the neuron activates for all the object samples while the minimum value is 0 (white) meaning the neuron never activates for this object. The figures represent that the number of activating neurons is higher the NeAW learning than the activity-agnostic NeAW-H and NeAW- aH learning, and more importantly, the neurons selectively activate depending on the object labels. Similarly, Figure[T]]shows the average activation in Hebb's rule, Grossberg' rule, and Oja's rule. As view the neuron activation in WTA modules as the cluster centroid, it means Hebb's 's rule creates many cluster centroids that group all the objects regardless of their labels, thereby, decreasing the dissimilarity between the objects in the latent space. In other words, it is hard to distinguish the rows (i.e. encoded vectors) in Figure 12(a). This is the reason why Hebb's rule gives a poor performance while having the balanced activity (Figure 9). Though Grossberg's rule and Oja's rule in Figure 12(b) and (c) display sparse columns and more object-dependent activation, the number of activating neurons small (~~3).. In this case, they create separable cluster centroids but the number of them are too few.
D IMPACT OF HYPERPARAMETERS IN NEAW LEARNING
We evaluate the impact of hyperparameters a and b in 3 with the variance of the neuron activity. Figure a3 and (c) show that the higher and b results in faster increase of the variance during training. This can be also understood through the geometric condition given in Theorem 1:
| W of (1)
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which describes that NeAW Hebbian learning relieves the biased activity only if the ratio of the Euclidean distance are lower than the ratio of (1 + 2) and (1 ). Given a and b can be regarded as different learning rates 17, higher and higher will increase the numerator and decrease the denominator, respectively. It results in the geometric condition for relieving the biased activity to become weaker, and consequently, such pair of a and b quickly increases the variance. However, in Figure b))b) and (d), we observe the maximum accuracy in different and b values are saturated to the similar level, which implies that the final accuracy with the NeAW Hebbian learning is not sensitive to and b.
a=0 or b=0 cases indicate that the either Hebbian or anti-Hebbian learning is used rather than the hybridization. In the very low rates such as 0, 0.01, 0.05, and 0.1 for a and b, the saturate accuracy and variance is still at the similar level. However, it is important to note that the variance of the neuron activity slowly increases in the lower rate as shown in Figure (3 and (c), indicating the
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Figure 13: Effect of and b for ModeINet4O. (a) the variance of the neuron activity with fixed and variable b, (b) the accuracy with the variable b values, (c) the variance of the neuron activity with variable and fixed b, and (d) the accuracy with the variable values.
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slow relaxation of the biased activity. can intuitively understood by the vector schematic in Figure 14] Though the NeAW learning with either a=0 or b=0 will not guarantee the fast relaxation of the biased activity, the direction of the weight update can still properly designed. Figure [5] shows the test accuracy for ModelNet40 with the models which encoder is only trained by 5 epochs. We observe that having higher a and b generally achieves higher accuracy than having very low and b values.
PERFORMANCE ANALYSIS IN VARIOUS MODEL ARCHITECTURE
Relation between Model Complexity and Performance We explore the smaller and larger model architectures with the supervised and unsupervised learning to understand how the performance changes depending on the model complexity. We design few simpler models with single-layer en- coder and classifier. Figure 16 shows how the test accuracy for ModelNet40 changes in the various models using end-to-end backpropagation and NeAW Hebbian learning. We observe the accuracy of the simpler supervised models clearly drops to 46.6%~61.9 perform better showing 56.1 1%.52.5% accuracy. The performance difference particularly large at the model which size 0.09MB: the unsupervised model achieves 71.7% and the supervised model
while the unsupervised models
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Figure 17: Self-supervised model architecture. reconstruction decoder with deconvolution and upsample layers is added on the proposed supervised and unsupervised model architecture.
Self-supervised Model and its Performance We design self-supervised learning model based on the proposed model Table 2. Figure 17 shows the schematic of the model architecture. The self-supervised learning cannot be directly applied on the same architecture and requires another autoencoder-basse decoder reconstruct the input point cloud. Hence, simple decoder with three deconvolution and upsample layers is added in the proposed architecture. The self-supervised model first trained with the encoder and reconstruction decoder to learn the latent representation. The main difference in training the self-superviser model is loss function. We Chamfer distance loss to compare the distance between ground truth and reconstructed point cloud. After the training, the parameters of the encoder fixed, and the classifier decoder trained by supervised learn-
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ing. We observe the self-supervised model achieves the similar accuracy in both ModelNet10 and ModelNet40 datasets with the supervised and unsupervised models. The accuracy is 92.2% for ModelNet10 and 79.5% for ModelNet40. However, we would like to note the computational cost of the reconstruction decoder. While the encoder and classifier used the inference are same with the supervised and unsupervised models (3. MBB), the number of parameters in the simple recon- struction decoder is 263,539 (1.0MB, total 4. 1.B... Also, during the training, the activation size per sample increases from 808 12, due the reconstruction. In other words, the computational cost for the inference is same, but the overhead of training is larger in the self-supervised learning.
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Figure 19: (a) Examples of point MNIST datasets and (b) test accuracy for point MNIST in super- vised learning (Backpropagation) and NeAW Hebbian learning.
Model Performance without WTA modules We would like note that the learning rule in Equa- tion (1) includes the indicator function. It represents that the weight update is based on the closest input point that is another WTA operation couple with the WTA modules. Hence, the proposed learning rule is designed along with the WTA modules, which is an important contribution in the paper. Figure [8] shows the test accuracy comparison between the model with and without the WTA modules in both ModelNet10 and ModelNer40. The model without the WTA modules achieves 71.8% for ModelNet10 and 25.5% for ModelNet40 while having the WTA modules shows 88.2% for ModelNet10 and 76.2% for ModelNet40. The results show that the model performance sig- nificantly drops without the WTA modules. Thus, this ablation study demonstrate that the NeAW leamiing should accompany the WTA modules.
Model Performance in Point MNIST We train and evaluate our supervised and unsupervised models on simple 2D classification task using point MNIST datasets. The model architecture is same with the other experiments. Each data samples include the point cloud representation of the MNIST images where the coordinates of the dark pixels are included. Figure 19 shows that the end-to-end supervised learning achieves 96.3%, and the NeAW learning shows the 94.8%. As the datasets relatively simple than ModelNet10 and MedelNet40, both the learning methods shows the similarly high accuracy. However, we observe that Hebb's 's rule, Grossberg 's rule, and Oja's 's rule achieve 58. 7%, 55.2%, and 69.0% test accuracy, respectively.
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Figure 20: Visualization of weight vectors. (top) the weight vectors in the last layer trained by NeAW Hebbian learning and (bottom) plain Hebb's rule. Dark elements refer the low values while bright elements are with the high values.
F VISUALIZATION OF WEIGHT VECTOR
Figure shows the weight vectors of the last layer in the encoder trained by different unsupervised learning rules. While plain Hebb's 's rule in Table achieves the higher variance of the neuron activity than the other learning rules, the variance between context vectors of different objects is low. indicates that the trained model different subset neurons for different objects. Given plain Hebb's rule repeatedly strengthens the weight of frequently activated pre- and post- synaptic neurons, the weight vectors will mostly learn the commonly appeared position vectors of objects, which degrade the sample-to-sample variation. Figure 20 describes the sparse weight matrix trained by Hebb's rule showing that the only few elements are with high values.
neurons represent objects, but it fails to activate the
G BIOLOGICAL PLAUSIBILITY OF NEAW LEARNING
As the key feature of NeAW Hebbian learning is to alleviate the biased activity issue, the question whether our brains have similar functions. We view the NeAW Hebbian learning is analogous to Homeostatic plasticity in biological brain (Turrigiano & Nelson 2004). It explains the exci- tatory and inhibitory neurons are dynamically controlled to relieve the biased activity and help the proper brain function such as memory (Keck et al. (017). It is still active research area, but one mechanism in such neurons to strengthen the inhibitory neurons onto excitatory neurons to reduce the activity, or vice versa. Our NeAW learning rule, which dynamically utilizes Hebbian learning (excitation) and anti-Hebbian learning (inhibition) as a function of the neuron activity, is similar with such neuron dynamics driven by the homeostatic plasticity in the brain. Also, NeAW Heb- bian learning is still locally defined unsupervised rule, which is an important feature of biologically plausible learning. However, according Dale's principle, the mature neuron releases the single type of neurotransmitters indicating the neuron can be either excitatory or inhibitory, while NeAW Hebbian learning assumes the neuron can be both the excitatory and inhibitory but function of the activity.
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Reviews:
[{"review_id": "fPC_ZOLim", "reviewer": "Reviewer_Sia3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors proposed an unsupervised learning method NeAW Hebbian learning that dynamically switches the learning rule of synapses between Hebbian (H) and anti-Hebbian (aH) depending on the activity of the output neuron. The proposed method is claimed to have reduced neuron activity bias, which leads to the activation of a different subset of neurons for different objects, thus, facilitate better learning. The authors tested the learning algorithm on 3D objects / point cloud recognition tasks and found that compared to Hebbian and anti-Hebbian learning, NeAW training reduces the activity bias of output neurons. On the 3D classification tasks, NeAW Hebbian learning rule also outperforms the existing Hebbian-based rules and end-to-end supervised learning when the data is limited. ", "summary_of_the_review": "This paper proposes a variant of the Hebbian learning rule which uses the output activity to decide if Hebbian or anti-Hebbian learning should be used. The proposed method claimed to have relieved the biased activity problem from existing Hebbian and anti-Hebbian learning rules, thus resulting in better local feature representation for downstream tasks such as 3D object recognition. It needs further validation to be a general unsupervised learning rule.  ", "strength_and_weaknesses": "**Strength**:\n\nIn general, this paper proposed an elegant extension of the Grossberg’s rule to address the biased activity problem for both Hebbian and anti-Hebbian learning rules. Some of the strengths of this work are listed below:\n\n- To facilitate the study of different Hebbian learning rules for 3D object classification, the author proposed a model with an encoder containing WTA modules at each layer. \n\n- The author identified the problem of biased activity (a few neurons are activated most of the time regardless of the object) with both Hebbian and anti-Hebbian learning in 3D object classification with their proposed model and linked this observation to the loss of local features in complex 3D objection recognition tasks. \n\n- To remove this bias, NeAW uses Hebbian rule when the output neuron activity is smaller than the average neuron activity p*=1/#output-neurons and uses anti-Hebbian when the activity is larger than p*. Intuitively, this seems like an implicit regularizor to reduce the activities of more activated neurons and increase the activity of less activated ones, and the activities are more evenly distributed across output neurons of a specific layer. \n\n- To illustrate this idea, the authors provided intuitive illustration to understand the geometry of the problem that NeAW is trying to address (Figure 3). And they also claimed to have proved NeAW relieves the biased activity under a given geometric condition. \n\n- The authors also empirically showed that the proposed NeAW results in a better feature representation which leads to better performance compared to other Hebbian-based rules and can even outperform end-to-end supervised learning when data is limited. \n\n**Weaknesses**\n\n- In the motivation study, the authors identified the biased activity problem with Hebbian-based learning rules and argued that a relaxation of this skewed activity is the key to improve learning. In addition to observing the activities resulting from Hebbian learning, I would also suggest studying if end-to-end learning with better classification performance would result in similar biased activities. \n\n- In this work, the proposed NeAW method is only studied with a specific task (3D learning) and a specific model. To make this learning rule more general, the authors are suggested to also test it on other tasks and model architectures. For example, would it also improve models without the WTA module?\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "YcbBaHLi_Z", "reviewer": "Reviewer_rCep", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents an unsupervised method to train deep neural networks using Hebbian/Anti-hebbian learning on 3D object classification from point clouds. The authors highlight (empirically and theoreticalaly) key disadvantages of using just Hebbian or Anti-hebbian learning; they demonstrate the improved expressivity and representational characteristics of using a neural activity based combination of both Hebbian and Anti-hebbian learning. They show that the latter outperforms other Hebbian learning rules on ModelNet10 and ModelNet40 3D object classification tasks.", "summary_of_the_review": "The paper presents an interesting combination of local learning with Hebbian/Anti-hebbian rules with deep neural networks and demonstrates its utility on 3D object classification on ModelNet10 and ModelNet40. The paper makes novel contributions that overcome key limitations of using Hebbian and Anti-hebbian learning rules in the context of deep learning -- prior computational modeling work has however experimented the combined use of Hebbian and Anti-hebbian learning rules (authors cite relevant work in this direction in Sec 3.3 Theorem 2). \n\nI have two main concerns: 1) Comparison to the current supervised techniques in the paper don't match model size, 2) there are other supervised methods that use very few parameters and achieve high performance on ModelNet40. That said, NeAW makes a significant improvement in using $\\textbf{local learning}$ to train deep neural networks. I recommend accept (6/10) at this stage due to the above two evaluation concerns. If the authors could show how their model performs when they increase model size, I may update my score as the current evaluation only uses a small number of parameters with model size 3.1 MB.", "strength_and_weaknesses": "Strengths:\n1. The authors present a neat combination of a biologically plausible local learning method (Hebbian/Anti-Hebbian networks are backed both by prior computational neuroscience modeling and experimental data on cortico-cortical connections) with deep neural networks for 3D object classification. The proposed NeAW Hebbian learning is clearly more expressive in the low-training data regime as shown in Fig. 7 even in comparison with supervised learning, this is very promising. \n\n2. NeAW quite significantly improves the performance of other Hebbian learning based methods. It is very interesting that this simple yet effective twist on Grossberg's rule results in ~15% and ~20% gains on ModelNet10 and ModelNet40 respectively.\n\n3. This paper is written really well with adequate graphical explanations to distinguish the different Hebbian learning rules from NeAW (Fig. 3 clearly shows how they are different from Hebbian and Anti-hebbian learning). Section 3.3 which is an important part of the paper that introduces NeAW is written particularly well.\n\nWeaknesses:\n1. Performance differences between NeAW and supervised methods on the full training set is quite big. Part of this can be explained by the difference in number of parameters between these models (model size). I think it would be fairer to compare NeAW when its model size is increased to match that of PointNet, PointNet++, DGCNN. The paper's contribution would seem even stronger if they match or improve performance of their unsupervised local learning rule with above mentioned supervised counterparts.\n2. There seem to be multiple supervised approaches I find at https://paperswithcode.com/sota/3d-point-cloud-classification-on-modelnet40 that achieve >90% on ModelNet40 with as few as 1.1M parameters (~4.4 MB if using fp32). If the authors would like to emphasize the strength of their model with few parameters, they must compare to these methods using very few parameters and yet achieving high performance on ModelNet.\n3. This is not necessarily a weakness per se, but I find it odd that the performance is insensitive to choice of a/b. Is it possible that the authors didn't cover a wide enough range to start seeing differences? Given that purely Hebbian or Anti-Hebbian methods work much poorly, setting very low learning rates for either (a or b) might produce significant performance differences. It would be great if the authors could test a wider range of a/b ratio and see if performance differences arise.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "WaRk8EmyJbe", "reviewer": "Reviewer_oMed", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes an unsupervised learning rule, NeAW, combining Hebbian and anti-Hebbian learning and applies it to 3D point cloud classification. The learning rule is motivated by the observation that Hebbian and anti-Hebbian learning don't produce diverse representations across neurons (there is low variance in neuron activity). The authors find that NeAW outperforms Hebbian and anti-Hebbian learning, achieves high performance with fewer parameters than comparable models, and outperforms backprop in low-data settings.", "summary_of_the_review": "The paper proposes an interesting Hebbian learning rule with some solid theoretical analysis. However, there are some concerns with the experiments and especially with the motivation of the paper as noted above. It would also be great if the authors could further clarify the novelty of the proposed rule.", "strength_and_weaknesses": "**Strengths**\nThe learning rule proposed by the authors is interesting, and some theoretical properties about it are proven. In my view, the most compelling point of this paper is the design of a learning rule that can produce strong representations of the data in an unsupervised way (as illustrated by Figures 4 and 5). Moreover, the theoretical analysis appears solid. A number of experiments are presented showing the properties of NeAW. The paper is also generally well written and easy to understand.  \n\n**Weaknesses**\nThe motivation for the paper is unfortunately a little unclear to me. What is the purpose of the NeAW rule applied to the 3D point cloud task? Do the authors wish to show good performance on the 3D point cloud task under model size constraints (as suggested by Table 2)? Do the authors wish to show good performance in low-data settings (as suggested by Figure 7)? Alternatively, is the main motivation to demonstrate an unsupervised learning rule that can balance neuron activity (i.e. Figure 2), with the 3D point cloud task as a case study?\n\nIf the motivation is to solve the 3D point cloud task more effectively, I would suggest emphasizing the relative advantages of NeAW more clearly throughout the paper. If the motivation is to propose a better Hebbian learning rule, I would suggest focusing the discussion and experiments on other similar Hebbian rules to emphasize that NeAW is superior. I would also suggest commenting on the biological significance of the rule since Hebbian learning rules have traditionally been proposed as models of biological neuron learning (e.g. to what extent can NeAW be considered biologically plausible?).\n\nIn terms of the specific motivation of NeAW rule, section 3.2 suggests that the poor distribution of neuron activity in Hebbian and anti-Hebbian learning rules leads to poor performance, and that NeAW achieves better performance by creating a more uniform activity distribution. While it is clear that NeAW achieves a more uniform activity distribution and higher performance, the causal link between these two is not clear to me unfortunately. Can the authors demonstrate empirically that a more uniform activity distribution *causes* higher performance (right now only a correlation is demonstrated)?\n\nI also have some concerns regarding the experiments. From Table 2, the paper argues that NeAW achieves good performance despite a lower model size. However, the model size seems linked to the model architecture used in the paper and not NeAW. If the authors wish to show that NeAW can be used to train small sized models more effectively than comparable learning rules, they may want to apply all the learning rules in Table 2 to a single (small) model architecture and evaluate performance. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster