Text Classification with Born's Rule
Authors: Emanuele Guidotti
Alfio Ferrara
Venue: NeurIPS 2022 Accept (2022)
Keywords: text classification
quantum physics
quantum-inspired machine learning
explainable classification

Abstract: This paper presents a text classification algorithm inspired by the notion of superposition of states in quantum physics. By regarding text as a superposition of words, we derive the wave function of a document and we compute the transition probability of the document to a target class according to Born's rule. Two complementary implementations are presented. In the first one, wave functions are calculated explicitly. The second implementation embeds the classifier in a neural network architecture. Through analysis of three benchmark datasets, we illustrate several aspects of the proposed method, such as classification performance, explainability, and computational efficiency. These ideas are also applicable to non-textual data.

Text Classification with Born's Rule
Emanuele Guidotti
Institute of Financial Analysis
University of Neuchátel, Switzerland
mnaanueleguidotii@ume
Alfio Ferrara
Department of Computer Science and Data Science Research Center University Milan, Italy
affioferrara@unmilt
Abstract
This paper presents text classification algorithm inspired by the notion of superpo- sition of states quantum physics. By regarding text as superposition of words, we derive the wave function of document and we compute the transition probabil- ity the document to a target class according to Born's rule. Two complementary implementations are presented. In the first one, wave functions calculated explicitly. The second implementation embeds the classifier neural network architecture. Through analysis of three benchmark datasets, illustrate several aspects of the proposed method, such as classification performance, explainability, and computational efficiency. These ideas are also applicable to non-textual data.
Introduction
In quantum physics, the superposition principle the idea that system can be in multiple states at the same time (e.g., a cat is simultaneously alive and dead in Schrödinger's thought experiment [22]). Here we show that text can be formally treated as a quantum system in superposition of words. Our onperpositionno--wortd model is a bag-of-wond ooowwrdss model where each word is treated as a quantum state. By exploiting this representation, we develop a supervised classifier based key postulates of quantum mechanics, namely the Born rule [4].
The Born rule provides link between the mathematical formalism of quantum theory and experiment, and as such is almost single-handede responsible for practically all predictions of quantum physics.' In this paper, represent documents and classes as quantum objects and we compute the probability of document to collapse target class by applying the Born rule.
The paper is structured follows. Section presents the motivation of this work. Section introduces the notation and some preliminary notions in quantum mechanics. Section develops our classification algorithm. Section 5 embeds the classifier in neural network architecture. Section 6 presents our empirical results. Finally, Section gives our concluding remarks and discusses extensions to semantics and non-textual data.
To simplify extensions to this work, we implement our classification algorithm in scikit-learn [17] and we embed the classifier in neural network architecture using pytorch [16]. All the code is available at https: t/gbbhbb.ttttttt// com/ httm///uuottttbbnnnnneee
We refer the reader to [14] for introduction to the Born rule.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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2 Motivation
In its most general form, the no-free-lunch theorem [23] implies that only prior knowledge makes possible to generalize from the training examples to novel test examples. As quantum mechanics represents our understanding of nature at the deepest level [13], we wonder whether quantum- inspired machine learning may incorporate a fundamental form of prior knowledge. Our hope that such knowledge results in more efficient algorithms, with better generalization ability and shorter computational times. Such algorithms would be also beneficial in terms of explainability and interpretability [8], as their interpretation is immediately inherited from the physical model [18).
This work should not be confused with quantum computing, which harnesses the properties of quantum states to perform calculations [1]. or with quantum machine learning [3], which explores how to devise and implement quantum software that could enable machine learning that is faster than that of classical computers [6]. We refer our work as caantum-inspired machine learning, in that we use quantum theory to derive machine learning algorithms that can run on classical computers. Although this not the first time quantum theory applied machine learning (see e.g., (15]), this is, to our knowledge, the first step develop new general-purpose algorithm for supervised classification based on key postulates of quantum mechanics, namely the Born rule.
Here we focus on text classification, as the analogy with quantum systems is straightforward. However, the formalism we present general and the implementations we derive can be used in practice as purppss classifiers.
3 Notation
Let x be a feature vector with elements Iy for j I,....J. J. Let y be a probability vector with elements yı 2 0 for k 1,.., K, and such that Σx 1/ 1. Our goal is to learn function g such that y = g(x). Then, given a test instance x', we predict the probabilities y' = g(x') and select the class K* = argmax; V/2 for classification.
We start by giving some preliminary notions quantum mechanics.
Wave function. In quantum physics, a system is regarded as superposition of states (s) and, using Dirac's notation [7], it is represented by a wave function |ı5):
(+b) Σ wslss with V. C.
Born rule. In the Copenhagen interpretation, the modulus squared of the inner product is interpreted as the (unnormalized) probability of the wave function (よا) collapsing to a new wave function lip):
P( + v) 1(
Σ
where C. denotes the complex conjugate of s.. This is known as the Born rule, and it is one of the fundamental postulates of quantum mechanics.
Wave coefficients. From (1) and (2), we notice that the coefficient |w-]2 represents the (unnormal- ized) probability of the wave function (gb) to collapse in state |s):
P( + 5) = (s| 2 = 1t../2.
(3)
Classification Algorithm
Let the feature vector x contain only non-negative elements such that xj 2 for all (c.g.. word counts or tf-idf weights). We regard zj as the (unnormalized) probability of the data instance (e.g.. document) to collapse in the j-th feature (e.g., word). We represent the j-th feature quantum state
2 We refer the reader to [19] for an introduction to quantum mechanics and its interpretation.
We refer the reader to [2. 21, 27, 28] for quantum language models.

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(J), and we represent the data instance with superposition of states ( = Σ3 Ryy5)) According (3) we have 2j P(e j) 1 which implies the natural choice 1i VTz..
[1b) = Σ (ols) Σ vzyli)
(4)
In a similar way, we represent the k-th class with a wave function (sp(A) = EE ( (j), and we obtain the coefficients 3 by setting the transition probability from (so(k)) to li) equal to the conditional probability of feature j given class k which we write as PJIA- According to (3) we have Paι P(yl(k) - j) = 1lo, p2, which implies the natural choice s Vakk-
(p(())
E ( Σ Vanan
(5)
Finally, we obtain the classification probability yk by computing the probability of (b) to collapse in (x)))) By substituting (4) and (5) in (2), the unnormalized probabilities are:
Uk
P( + ( 10))1
Σ
p
(h) و3
(EvPiise PIINTT))
(6)
and the normalized probabilitiess are yk = uk/ Σx uk.-
4.1 Training
To obtain the conditional probability Pitk in (6) proceed follows. Given training set we normalize each feature vector x()) such that it sums up 1:
{(x(")
(n)
Σ y ạ
Then, we compute the conditional probability Patk from the (unnormalized) joint probability Pya:
Pzk
Σ
2j yk
(m)
(n) E, Pyr
PjAk
Pje
(8)
4.2 Regularization
We observe that if Pate is constant for k = 1, 1,..., K, then the j-th addend increases the summation in (6) by the same value for all classes. As un is a monotonic transformation of the summation, the j-th addend does not alter the ranking of the probabilities u,, thus being irrelevant for the final classification A* = argmax y = argmax; ug. To regularize the predictions, we shrink the contribution of irrelevant addends towards zero by re-weighting the summation in (6).
Let us rewrite, for case of notation, the probabilities Pjlk as some weights W,k 2 0. Then, j is irrelevant if WýR is constant for all k. We normalize Wia such that the weights of the classes & sum up to for each j. that is:
Wjk
WHI Ex Ws
An irrelevant maximizes the entropy Hy - Wa j), as WMi is uniformly distributed across the classes k. Thus, we introduce the following weights that range between 0 (irrelevant with maximum entropy) and (relevant j with null entropy):
H,
Σ. Way In(Wa))
(10)
Hmaz
In(E. 1)
Finally, we use H, in (10) to re-weight the summation in (6), which becomes:
ug
Σ H Hj PJIET,)
(1I)
3

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4.3 Generalization
To simplify ablation studies, we generalize (11) as follows:
U Σ () with WiR- E Py P3x)
Psk
(12)
where H2 is given in (9)(11), Ps is given in (8), and a > 0, 2 0, and 2 0 are the model hyper-parameters. Here, we are mainly interested in the choice a = 2. b 1, and = 1, which corresponds to the original model in (11). Another special configuration is 1, 0, and h = 0, which corresponds to 4 = Σj PuIsT where Pklj is the conditional probability of & given j. This configuration offers a natural benchmark for our quantum approach in that it computes uy according to classical probability theory.
4.4 Explainability
The contribution of the j-th feature the total probability ur (local explanation) is given by the addend Wi 3k in (12). Therefore, the most influential feature for the classification ke is given by j argmax, Hf 5 23. In general, we use HJ) -x3- to rank the features by the degree in which they contribute to the classification k*.
The explanation the class level (global explanation) is obtained by investigating the product H Wa in (12), regardless of the vector x. The global most influential feature for each class is given by jk = argmax; H, W In general, we use H. W to rank the features by their global importance jk-
jk
with respect to class k:
4.5 Computational Complexity
In the training phase, the algorithm in (11), and more generally (12), learns the joint probability P5K in (8) by multiplying the x N matrix of elements zi with the N K matrix of elements yk Entry Pys is given by the inner product of the j-th row of the left matrix (which has N entries) and the k-th column of the right matrix (which has N entries), so computing it takes time ONN). We do this once per element. Since the output matrix has dimension J x K, there are O(JK) elements to consider and the total work is done in O(JJK) That is, the training time is at most linear in the number of samples (N), in the number of features (J), and in the number of classes (K).
(n)
In the prediction phase, we compute elementwise operations on the J K matrix of elements Pyk. the total work done time (JK)) That is, the prediction time is at most linear in the number of features (J), and in the number of classes (K), and it does not depend on the number of training samples (N).
(and yk.) in (11), and more generally (12). As these are all
Finally, we notice that the computational complexity can be further improved by using sparse matrices, and all the operations involved in the training and prediction phases can be easily parallelized (e.g.. on GPUS). Thus, we expect the method to be highly scalable.
5 Neural Architecture
A major limitation of the algorithm presented in Section 4 is that it can be applied only when 2y 0. Here we embed the method in a more flexible architecture that admits 2, E C.
Let us assume that data instance (e.g., document) can be represented as a superposition of some hidden states [s) for s = 1,..., S (e-g., word embeddings). Then, we write its wave function (15) = (slss) where the coefficients the generally depend on the feature vector x (e.g., words). We represent such coefficients with a neural network vi = +x((x) that maps the feature vector E Cd to the vector of wave coefficients vé E CS. Then, we write the wave function of class k as (o(()) E, (*) (s) where the coefficients xp!*) depend on & and s, but not on the feature vector x.
Finally, we use the Born rule in (2) to compute the probability of (1)) to collapse in (k))));
us = P(b + (+)) = -1(p(0)
Σ
c the (x))
2
(13)

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Equation (13) is read as neural network o( where d is matrix of elements t 5 v = (x) is the output of the previous layer, &v denotes the matrix product Σ, 2x. and the activation function (-)) -1- l is the modulus squared. Finally, we apply a normalization layer to obtain the probabilities yı: uz/ >x k uk E [0, 1]. An illustration is given in Figure 1.
Re(z)
u
X
(x)
tv
r² + 12
|
y
Im z
Figure I: Born Layer (BL) architecture.
Our method is similar to the usual approach where the first part of network is assumed to learn log pog-prbbabilitie, and the final classification layer converts them into probabilities by applying the softmax function. Here, the first part of the network is assumed learn the coefficients of wave function, and the final classification layer converts them into probabilities by applying the Born rule.
5.1 Initialization
We initialize the weights 2x. such that the wave function (k) has an equal probability to collapse in any state [s). To this end, let us write the complex-valued weights lks ehae "M,, where
is the imaginary unit, Piss 2 O, and Oz. E [0, 2T). We recall from (3) that the probability of
(A)) collapse in (s) is ((yo(A) the (unnormalized) probability is constant for all & and s. To normalize the probability, we set Sp² = where S is the number of states (input dimension of the layer). We obtain which resembles the initialization in [9] and [10]. Finally, we sample O2z from uniform distribution in the interval [0, 2x), such that the weights tx are uniformly distributed in the complex circle (isotropy).
1sp (1) k ph.. We set p p² so that
1/v5.
4 R.
with Oks ~ ~2000,
(14)
When the feature vector x (unnormalized) probability vector in Section 4, then (11) can be written as the neural network in (13) where we use 5 = and (x(()) = /z.. In this case, it is interesting to initialize the weights in (13) with the corresponding weights developed (11), that LR. H. ajkk"
5.2 Explainability
We notice that the probabilities yv are invariant under scaling and rotation of the coefficients so(+) in (13). To show that, we multiply & by scaling factor p and phase factor et Then, we substitute * + 2e p(h) in (13). The phase factor e0 vanishes when computing the modulus and the scaling factor vanishes when the probabilities are normalized.
The invariance by scaling and rotation implics that the weights 4 have absolute meaning, and they become meaningful only in relation with each other. To inspect the relations among states s, for given class k, it is interesting axes). Here, the length of the vectors can be used to rank the states by importance, while the direction of the vectors produce constructive destructive interference among the states (see Figure 5 for an illustration). For local explanations, PR.U. is used instead of &
visualize the weights Tk in the complex plane (with unlabelled
'More precisely, we scale the weights by dividing them by their mean and by the square root the number of features to mimic (14). This does not alter the model, which is invariant under scaling (see Section 5.2), but it helps prevent vanishing exploding gradients during backpropagation.
5

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Empirical Results
We illustrate several aspects of our classifier using three well-established text classification bench- marks: 20Newsgroup", and the R8 and R52 subsets of Reuters 21578 We perform tokenization using the function nltk. word pord__ooeeize and vectorize the text with Tfid/Vectorizer". No other text trans- formation cleaning procedure performed. The final datasets are composed by (20Newsgroup) 20 classes, 204 817 words, 11 314 training documents, and 7 532 test documents; (R8) 8 classes, 33 593 words, 5 485 training documents, 2 189 test documents; (R52) 52 classes, 38 132 words, 6 532 training documents, and 2 568 test documents.
All the results are obtained using Python 3.9 Google Cloud Virtual Machine equipped with CentOS 7, 12 vCPU Intel Cascade Lake 85 GB RAM, GPU NVIDIA Tesla A100, and CUDA 11.5.
6.1 Training time, prediction time, and accuracy score
We compare our methodology against baseline of six classifiers on the 20Newsgroup dataset. With Born Classifier (BC), we refer to the algorithm presented Section 4, where weights are computed in equation (8) and classification probabilities are calculated in equation (11). The baseline is composed by the algorithms Decision Tree (DT), K-Nearest Neighbors (KNN), Random Forest (RF), Support Vector Machine (SVM), Multinomial Naive Bayes (MNB), and Logistic Regression (LR). For all the algorithms the baseline, we use the corresponding implementation in scikit-learn. All the classifiers executed CPU with default parameters.
BC
+
BC IGPU)
+ DT
KNN
LR
MNB
+
RF
+
SVM
0.9
Training Time (s]
Prediction Time (s)
107
Accuracy Score
0.8
10a
30a
10a
0.6
30'
0.5
10-2
10-
10
0.4
Dataset Size
Dataset Size
Dataset Size
(fraction of training data)
(fraction training data)
(fraction of training data)
Figure 2: From left right: training time, prediction time, and accuracy score the 20Newsgroup dataset (y-axis) for several classifiers, in function of the fraction of data used for training (x-axis).
The comparison between BC and the baseline is reported in Figure 2. where we show the training time, prediction time, and accuracy score in function the fraction of data used for training in 10 independent executions. Figure shows that BC is fast to train, is fast to predict, can be accelerated on GPU, and it achieves the highest accuracy regardless of the size of the dataset. Moreover, as the amount of training data decreases, the accuracy gap between BC and the other classifiers widens.
6.2 Imbalanced data
While 20Newsgroup almost balanced in terms of documents per class, R8 and R52 are not. In particular, the most frequent class in R52 contains 2 840 training samples, while the least frequent class contains only document in the training set. Figure 3 reports the F1-macro score for BC and the baseline the three datasets, which models and the performance gap widens for more imbalanced data. The native capability of BC to work with imbalanced data can be traced back to (8), which computes the conditional probability of the features given the classes. Dividing the joint probability by the marginal effectively normalizes by the class imbalance.
increasingly imbalanced. BC outperforms the bascline
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BC
DT
KNN
LR
MNB
RF
SVM
0.8
F1-Macro Score
F1-Macro Score
0.6
0.6
0.4
0.3
0.4
0.2
0.3
0.2
0.9
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Dataset
Dataset Size
Dataset Size
(fraction of training data, 20NG)
(fraction of training data, R8)
(fraction of training data, R52)
Figure 3: FI-macro score (y-axis) on 20Newsgroup, R8, R52, for several classifiers, in function of the fraction of data used for training (x-axis).
6.3 Hyper-parametert and ablation study
We now compare the overall accuracy score and runtime of BC with the results of the fine-tuned baseline on 20Newsgroup (see Table 1). We then conduct an ablation study on BC (see Figure 4).
0.8
Model
Accuracy (%)
Runtime (s)
DT
53.9
5,583.
0.6
KNN
55.6
640.574
RF
77.5
49,686.936
0.5
1.0
SVM
79.4
45,639.071
0.5
0.4
LR
82.9
966
0.0
MNB
84.1
15.320
1.5
BC (11)
87.3
0.043
0.0
1.0
0.5
0.1
Table 1: Accuracy score and runtime for several classifiers the 20Newsgroup dataset. The run- time is the (CPU) time to optimize the model's hyper-parameters by 5-fold cross-validated grid- search on the training set, plus the time to refit the selected model. The accuracy score is the accuracy achieved by the model on the test set.
Figure 4: Ablation study on the 20Newsgroup dataset. The figure displays the test accuracy
scores of the model in (12) for several val- ues of the hyper-parameters a, b, and h. The biggest point identifies the configuration of hyper- 1, which corre- sponds to the original model in (11).
parameters a 1.
In Table 1, we tune the baseline classifiers via grid-search using 5-fold cross- wooss-vliiattin. Except for BC, which needs no tuning, all the other classifiers use between 20 and 50 combinations of hyper-parameters (reported in the replication code). BC still achieves the highest accuracy by remarkable margin. Furthermore, the overall time required to classify the dataset is several orders of magnitude smaller than than that required by baseline algorithms. This partly due to the speed of execution of BC (see Figure 2) and partly due to the fact that BC has no parameerssssters tune.
The ablation study is reported in Figure 4. Here we display the test accuracy scores of the model corresponding to equation (12) for several values of the hyper-parameters a, b, and h. The config- h 1, corresponds to BC in equation (11) and it achieves 87.3%. Selecting the hyyper-paramedd suggesting that our prior generalizes well. The classical configuration in Section 4.3, achieves an accuracy of 83.3%. Removing the regularization (h = 0) from our model leads to a large drop in performance: from 87.3% to 77.7%. On the other hand, adding the regularization (h = 1) to the classical configuration does not lead to any performance gain: from 83.3% to 82.0 0%. These results can be understood by recalling that the classical configuration uses the conditional probability of the classes given the features (Pei)), while BC uses the conditional
uration 5. b
accuracy of via cross validation also achieves a test accuracy of 87. 73,%
1, 0, h 0. discussed

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probability of the features given the classes (P As Pj gives larger weight to more frequent features, then BC is dominated by noise when the features irrelevant for classification are also the most frequent (as is typical in text). Regularization becomes important to reduce the influence of frequent features that are slightly skewed. Instead, in the classical configuration, regularization redundant Pay already limits the influence of such features.
6.4 Embedding in a neural network architecture
With Born Layer (BL), we refer to the neural architecture displayed in Figure that takes in input (x(x) and returns the classification probabilities according to equation (13). Here we choose (x(()) vTs so that the transformation in equations (11) and (13) are the same, and they can be directly compared (in general, (xx) may be an arbitrary transformation to exploit e.g., semantics, as discussed in Section 7). BL is initialized as in (14) and it is optimized on the L'-loss using Adam [12] with default parameters. We also initialize BL with the weights computed by BC as discussed at the end of Section 5.1. We refer to this version with BC+BL.
We stress-test BC and BL (which only rely on word counts) against a variety of deep learning approaches reported in literature (which exploit more sophisticated representations of the document). In Table 2, we include cooperative neural networks (CoNN) [20], models that learn distributed representations of entities and documents from knowledge base (TextEnt) [25], graph convolutional networks (TextGCN) [20], the neural attentive bag-of-entitee model (NABoE) [24], and the diversified ensemble neural network (DEns) [29].
We find that BC achieves ofate-of-the-an accuracy on the 20Newsgroup dataset, and it is up million times faster than alternative approaches (0.001s). This confirms that the weights computed explicitly in (11) are good prior. Also BL matches state-of-the-ate performance, but it requires significantly longer computational times (about 34s). Initializing BL with BC requires few training epochs to further improve performance, suggesting that BC+BL can be fine-tuned relatively fast (about 3s). Finally, we notice that, while BL optimizes complex-valued weights, BC+BL uses real-valued weights. This suggests that the reason for good performance lies in the transformation itself (13), rather than in the usage of complex numbers.
Accuracy (%)
Runtime (s)
20NG
R8
R52
20NG
R8
R52
CoNN [20]
83.7
N/A
N/A
120.000
N/A
N/A
TextEnt [25]
84.5
96.7
N/A
923.089
556.020
N/A
TextGCN [26]
86.3
97.1
93.6
1206.372
109.184
186.531
NABoE [24]
86.8
97.1
N/A
152.154
24.110
N/A
DEns [29]
87.1
97.7
94.3
N/A
N/A
N/A
BC (11)
87.3
95.4
88.0
0.001
0.001
0.001
BL (1 epoch)
84.6
96.5
87.9
0.347
0.276
0.274
BL (10 epochs)
86.2
96.8
92.6
3.451
2.747
2.723
BL (100 epochs)
87.1
97.1
92.7
34.461
27,452
27.171
BC+BL (1 epoch)
86.9
97.5
91.8
0.348
0.278
0.276
BC+BL (10 epochs)
87.4
97.7
95.2
3.458
2.764
2,724
BC+BL (100 epochs)
87.4
97.2
94.4
34.521
27.494
27.124
Table Accuracy score and runtime for several methods on the 20Newsgourp, R8, and R52 datasets. The runtime is the (GPU) time to train the model. The accuracy score is the average score achieved by the model on the test in 10 independent runs. Standard deviations are below 0.1 (%) and are omitted from the table. The runtime for CoNN is taken from the original paper [20]. The runtime for the other methods are obtained by running the corresponding code on our hardware infrastructure.

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6.5 Explanation
To exemplify the explanation provided by BC, we extract the global weights described in Section 4.4 and we report the 10 most influential words (out of more than 200 000) for subset classes in 20Newsgroup. Table 3 shows that the words do not contain stopwords or punctuation, or terms that are too general to be representative of a specific class. The ability of BC to filter out noise can be traced back to the regularization in Section 4.2 that shrinks to zero the weights of words with little discriminatory power:
#
Baseball
Hockey
Autos
Graphics
Macintosh
Windows
Cryptography
Phillies
NHI.
polygon
Centris
AA
encryption
2
Braves
hockey
cars
TIFF
Quadra
Windows
Clipper
3
pitching
Leaf's
eliot
graphics
Apple
3.1
clipper
4
Alomar
team
SHO
3D
Mac
windows
crypto
5
Baseball
Devils
automotive
3DO
Duo
W4WG
NSA
6
Players
ESPN
Callison
CView
LCIII
cica
escrow
Mets
Wings
Dumbest
POV
LC
font
key
8
Sox
Pens
rmt6r
cview
C650
BJ-200
DES
Cubs
playoffs
Thigpen
tdawson
BMUG
NDIS
Amanda
10
baseball
playoff
Toyota
MPEG
ITsi
Win
wiretap
Table Global explanation on 20Newsgroup as described in Section 4.4. The table shows the top 10 features (words) for the classes: Baseball boc.sppot..baeball) Hockey (rec.sport.hocke)) Autos (rec.autos), Graphics (comp.graphicss Macintosh soomp.sys.mac.hare Windows (comp.os.ms windows.misc) Cryptography (sci.crypt).
In Figure 5, we train BL for two epochs and represent the evolution of the complex-valued weights described in Section 5.2. The figure represents the 10 weights that have the largest (and smallest) modulus for the class woo.sport.basebaa initialized randomly in all directions of the complex circle, BL learns to give larger weights to more relevant features, smaller weights to less relevant features, and orientate the features in different directions. The weights converge in the same direction for words that are specifically related to the concept of baseball. The words hockey and NHL (National Hockey League), which are still relevant for different class of sports, are associated with a large weight but they are also collocated in the opposite direction with respect to the baseball-related words.
the end of the training. Starting from weights that
Sostal MH..
Mela
590 upccet
phaching
thilles
tockey
Nittr
-96- 55c50
-90 OP
40- Drocula
-0-- reedless
Initial Weights
Epoch
Epoch
Figure 5: The figure shows the 10 most influential features (solid lines) and the 10 least influential features (dotted lines) for the class rec. rec.sport. baseball in the 20Newsgroup dataset, as described in Section 5.2. The evolution of the (complex-valued) weights is represented from left to right.

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7 Conclusion
By regarding text as a superposition of words, we have introduced a supervised classification algorithm based on key postulates of quantum mechanics, namely the Born rule. The classifier is self-explainable, admits a closed-form expression (II), and can be embedded in neural network architectures (13). Our method outperforms traditional algorithms the 20Newsgroup, R8, and R52 datasets, terms of classification performance and computational time. These results suggest that physical principles can be successfully exploited in machine learning and may open new class of classification algorithms.
There are several potential improvements and extensions to this work. First, our classifier is derived from the Born rule and the explanation for its effectiveness in the view of machine learning remains an open question. Improving the empirical evaluation with thorough experiments would allow to assess the limitations of the method and describe the benefits and pitfalls in various applications and data types. Second, whenever data instance can be represented with a probability distribution over some space, then a wave function can be constructed explicitly and the data may be classified with the algorithm in Section 4. More generally, network in Section 5, where the first part of the network learns the coefficients a wave function and the final classification layer converts them into probabilities by applying the Born rule. Using the transformation in (13) on top of cxisting architectures offers a natural way to push our framework beyond a bag-of- worddds
arbitrary data instance can be classified with the
representation of the document, to encode semantics, and to extend the method to non-textual data. For instance, an image may be classified with the network in (13) where (x) is convolutional layer. This corresponds regarding the image superposition of the features learnt by the convolution. Finally, an interesting research direction would be constructing deep networks that apply the transformation in (13) repeatedly. A deep architecture of this kind would find its biological foundations in the quantum brain hypothesis [11], which suggests that quantum events could play a non-trivial role in neuronal cells, and contribute an extremely high complexity, variability and computational power of neuronal dynamics.
Acknowledgments
This work was partially supported by the Google Cloud Research Credits program with the award GCP19980904.
References
[1] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 57407779::0055510, 2019.
[2] Ivano Basile and Fabio Tamburini. Towards quantum language models. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 1840-1849, 2017.
[3] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549(7671):115-22 202, 2017.
[4] Max Born. Quantenmechanik der stoßvorgänge. Zeitschrift für Physik, 38(11):803-82) 1926.
[5] Yiwei Chen, Yu Pan, and Daoyi Dong. Quantum language model with entanglement embedding for question answering. JEEE Transactions on Cybernetics, 2021.
[6] Carlo Ciliberto, Mark Herbster, Alessandro Davide Ialongo. Massimiliano Pontil, Andrea Rocchetto, Simone Severini, and Leonard Wossnig. Quantum machine learning: A classical perspective. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 774(2209):20170513 2018
[7] Paul Adrien Maurice Dirac. A new notation for quantum mechanics. In Mathematical Proceed- ings of the Cambridge Philosophical Society, volume 35, pages 416-418. Cambridge University Press, 1939.
10

---

[8] Filip Karlo Došilović. Mario Brčić, and Nikica Hlupić. Explainable artificial intelligence: A survey. In 2018 s1st Interaational Convention on Information and Communication Technology. Electronics and Microelectronics, pages 0210-0215, 2018.
[9] Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 249-256, 2010.
[10] Kaiming He. Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE International Conference on Computer Vision, pages 1026-1034, 2015.
[11] Peter Jedlicka. Revisiting the quantum brain hypothesis: Toward quantum (neuro) biology? Frontiers in Molecular Neuroscience, 10:366, 2017.
[12] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
[13] Daniel Kleppner and Roman Jackiw. One hundred years of quantum physics. Science, 28955481) 2000.
[14] Nicolaas P Landsman. Born rule and its interpretation. In Compendium of Quantumm Physics, pages 64-70. Springer, 2009.
[15] Qiuchi Li, Dimitris Gkoumas, Alessandro Sordoni, Jian-Yun Nie, and Massimo Melucci. Quantum-inspired neural network for conversational emotion recognition. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 13270-13278, 2021.
[16] Adam Paszke, Sam Gross, Franeisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems, volume 32, 2019.
[17] Fabian Pedregosa, Gaäl Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vander- plas, Alexandre Passos, David Couraapaau, Matthieu Brucher, Matthieu Perrot, and Edouard Duchesnay. Scikit-leam: Machine learning in python. Journal of Machine Learning Research, 12(85):2825-28300 2011.
[18] Carlo Rovelli. Relational quantum mechanics. International Journal of Theoretical Physics, 3578):1637-1678 1996.
[19] Carlo Rovelli. Helgoland. Prometheus, 2021.
[20] Harsh Shrivastava, Eugene Bart, Bob Price, Hanjum Dai, Bo Dai, and Srinivas Aluru. Coopera- tive neural networks: Exploiting prior independence structure for improved classification. In Advances in Neural Information Processing Systems, volume 31, pages 4126 4136, 2018.
[21] Alessandro Sordoni, Jian-Yun Nie, and Yoshua Bengio. Modeling term dependencies with quantum language models for IR. In Proceedings of the 36th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 653-662, 2013.
[22] John D Trimmer. The present situation in quantum mechanies: translation of Schrödinger's "cat paradox" paper. Proceedings of the American Philosophical Society, pages 323-338, 1980.
[23] David H Wolpert and William G Macready. No free lunch theorems for optimization. JEEE Transactions on Evolutionary Computation, 1(1):67-82, 1997.
[24] Ikuya Yamada and Hiroyuki Shindo. Neural attentive bag-of-entities model for text classification. In Proceedings of the 23rd Conference on Computational Natural Language Learning, pages 563-573, 2019.
[25] Ikuya Yamada, Hiroyuki Shindo, and Yoshiyasu Takefuji. Representation learning of entities and documents from knowledge base descriptions. Proceedings of the 27th International Conference Computational Linguistics, pages 190-201, 2018
11

---

[26] Liang Yao, Chengsheng Mao, and Yuan Luo. Graph convolutional networks for text classifi- cation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 1370-7377,2011
[27] Peng Zhang, Jiabin Niu, Zhan Su, Benyou Wang, Liqun Ma, and Dawei Song. End-to anddto-ennd quantum-like language models with application question answering. In Proceedings of the AAAI Conference on Artificial Intelligence, 2018.
(28) Peng Zhang, Zhan Su, Lipeng Zhang, Benyou Wang, and Dawei Song. A quantum many- body wave function inspired language modeling approach. In Proceedings of the 27th ACM International Conference Information and Knowledge Management, pages 1303-1312, 2018.
[29] Shaofeng Zhang, Meng Liu, and Junchi Yan. The diversified ensemble neural network. In Advances Neural Information Processing Systems, volume 33, pages 16001-16011, 2020.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Section
(c) Did you discuss any potential negative societal impacts of your work? [N/A] We don't ' any potential negative societal impact other than those that apply to ML in general.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as URL)? [Yes] The code to reproduce the results is provided in the supplemental material.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
4. If you are using existing assets (c.g., code, data, models) or curating/relcasing new assets...
(a) If your work uses existing assets, did you eite the creators? [Yes] Datasets (2ONG, R8, R52) and software (scikit-learn, pytorch) are cited properly.
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any assets either in the supplemental material or as a URL? [Yes]
(d) Did you discuss whether and how consent obtained from people whose data you' 're using/curating? [N/A]
(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
12


Reviews:
[{"review_id": "do1U_Defjd", "reviewer": "Reviewer_Jcpw", "summary": "The paper describes an interesting application of Born Rule to one of the most fundamental tasks in machine learning which is text classification. The Born Rule is a concept in quantum physics on estimating the probability of an object (or document in the context of this study) to collapse in one of the probable outcomes (category/class of the text). In this line, the paper frames text classification as a superposition of words where Born Rule can be integrated as a standalone classifier or with integration to a neural classifier. The authors show that the method outperform both traditional and neural network based methods on three datasets (NG, R8, and R52) by a considerable amount of margin especially with >1 epochs. There is also an emphasis on the efficient runtime obtained by using the proposed method with Born’s rule. ", "questions": "1. What was the motivation for traditional algorithms when the use of language models such as BERT are already considered baseline in text classification as they are equally accessible compared to the tested algorithms? Also, what were the criteria for choosing the three datasets when IMDB, Amazon, Yelp, AG are often used for the evaluation of newly-proposed TC methods (see Patel et al, NAACL 2021).\n2. Are there any cases where TC with Born’s rule favors particular classes based on their confusion matrices? The paper only shows Accuracy scores so there is uncertainty if it resolves class imbalance or not. This can be a limitation of the method.\n", "limitations": "The paper covers a new method for implementing text classification that seems to outperform both traditional and some neural network-based methods in terms of speed and at par with accuracy scores. The authors are advised to exhaustively list what the proposed method is (not) capable of doing covering factors such as context, how texts is processed, type of task. Again, this can be described through a table. \n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. The paper is fairly easy to read, and follow, and contains an arguably sufficient level of technicality that would make even intermediate readers be interested in the concepts in quantum physics such as the Born rule. I also appreciate that the authors conveniently built the method on top of Scikit and Pytorch for ease of replication on the side of the readers. \n2. For this new approach, I expected a very definitive discussion on the limitations to describe the pitfalls of the proposed method in various applications and data types. Will it work on cases such as a sentiment classification task where some level of context is needed and there are cases of double negatives? Will it work on extremely noisy texts such as those collected from social media (Twitter)? What is the effect if we reduce the size of each text instance (say sentence classification instead of document-based classification)? \n3. Following #2, a definitive synthesis comparing TC with Born’s rule to existing methods of text classification (using language models, traditional method) should be in order. This can be tabulated for ease of reference.\n\nWith all the strengths and weaknesses of the paper, I believe it would still be beneficial for the ML community to learn from the paper provided the authors studiously highlight the limitations of the method. Thus, I would recommend acceptance on my part.\n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "MqgZgPaJIvJ", "reviewer": "Reviewer_5Hcj", "summary": "This paper proposes a novel algorithm derived from Born's Rule in quantum physics for text classification. The paper discusses various aspects of this classification algorithm, including training, regularization, generalization, explainability, and computational complexity. A neural architecture called Born Layer is also developed. The empirical results show the superiority of the proposed algorithm in prediction accuracy and computational complexity over several classical baseline ML algorithms. This work is promising to be the first step to develop a brand new general classification algorithm with certain explainability inspired by quantum physics.", "questions": "- In Figure 2, the training time of KNN increases as the dataset size grows. This implies that the compared KNN is not a brute-force one. Moreover, I would expect to see more details to be elaborated on for traditional algorithms in the supplemental material.\n- The Born Classifier shows a significant superiority over other classical algorithms when the dataset size is limited. This is surprising. Do you have any interpretation of this observation? Do you have such justification for other datasets?\n", "limitations": "The author does not discuss the limitations.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**\n\n- I love the idea of a classification algorithm based on the theory of quantum physics. The authors have a comprehensive discussion of various aspects of the proposed classification algorithm.\n- The algorithm is implemented with well-structured code, which should make the community easy to extend and try on more tasks.\n- The algorithm shows impressive superiority over baseline algorithms in the prediction accuracy and training/inference efficiency.\n\n**Weakness**\n\n- The empirical result is limited. The major experiments are only conducted on 20NewsGroup. Also, the compared baselines are relatively weak. (But considering this is a preliminary work in this direction, this is acceptable.)\n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "F7KtSNQIZp", "reviewer": "Reviewer_Sqxh", "summary": "\nThis paper uses the mathematical tools of quantum mechanics to realize a fast and accurate text classification algorithm. Specifically, in this paper, a document is regarded as the superposition state of words, and the classification is viewed as the document to collapse to another superposition state associated with the target class. The collapse probability of the document can be computed by the Born rule. Most of the experimental results are better than baseline models. In addition, the authors also give the interpretability analysis of the two methods. The interesting experimental results show that the classification algorithm is effective and interpretable.\n", "questions": "Q1: The motivation of this paper says the prior knowledge results in more efficient algorithms. What is prior knowledge in this work, and does it give an intuitive understanding at this point?\n\nQ2: The author says this work is the first supervised classification algorithm based on quantum theory. In my opinion, this view is not rigorous. Because many quantum-inspired models have been used to supervised classification tasks, such as QMNN[1], Complex-order[2]. In addition, there are some other quantum-inspired language approaches.e.g., quantum language models, quantum many-body wave language models. What is the advantage of this work over these language approaches?\n\nQ3: On line 68, $|j\\rangle$ is represented as word, and the document is represented as a superposition of state $|\\psi\\rangle = \\sum_j \\psi_j |j\\rangle$. In quantum theory, the superposition state indicates that a quantum state is superimposed by its eigenstates, and will collapse to an eigenstate after measurement. However, the superposition state represent a document, which means that a word will be used to represent the document after measurement. Even if there is no measurement and the document is expressed as the weighted sum of word vectors in the code, its physical meaning is still unreasonable. \n\nQ4: On line 71, author represents each class k with a wave function $|\\phi^{(k)}\\rangle = \\sum_j \\phi_j^{(k)} |j\\rangle$. Why the class k is the superposition state of word? Does this mean that class k will be represented by a word in the document after measurement?\n\nQ5: In equation 5, the $p_{j|k}$ means the conditional probability of j given k. It is necessary to specify the relationship between j and k, and the reason why the conditional probabilities are used as the superposition state coefficients.\n\nQ6: On line 80, the $x^{(n)}$ is feature vector. Is $x^{(n)}$ a word or $|j\\rangle$? What is the relationship between $y^{(n)}$ and $y_k^{(n)}$? And the $y_k^{(n)}$ has no definition.\n\n\nQ7: On line 85, author says “the j-th addend does not alter the ranking of the probabilities $u_k$, thus being irrelevant for the final classification”. However, each $\\sqrt{p_{j|k} x_j}$ contributes to $u_k$ in the equation 6. Why the j-th addend is irrelevant for the final classification? The reason why entropy is used as weight is not given. Can you introduce the rationality of entropy? In the equation 10, is $u_k$ a normalized vector?\n\nQ4: On line 136, $\\Phi$ is a matrix of weights， and $\\Phi_{ks}$ is a scalar. Why are they equal? Is that ambiguous? Similarly, $v=v_s$ is also confusing.\n\nQ6: Whether it can be understood in this way, the 145 to 152 line gives the initialization process of BL, and the 153 to 155 line gives the initialization process of BL by BC.\n\nQ7: On line 159, Why does $\\rho e^{i\\\\theta} \\overline{\\varphi}_s^{(k)}$ have to be derived in this form? Can't $\\overline{\\varphi}_s^{(k)}$ be directly interpreted?\n\nQ8: The novelty of this work is mainly Born rule, but most quantum-inspired models use quantum measurement theory to classify or judge. What is the essential difference between this work and other quantum-inspired models?\n\nQ9: The reasons for choosing these baseline models need to be explained. Is there comparability between this work and other baseline models in ideas, methods and other aspects? Why \n\n[1] Li Q, Gkoumas D, Sordoni A, et al. Quantum-inspired neural network for conversational emotion recognition[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2021, 35(15): 13270-13278.\n\n[2] Wang B, Zhao D, Lioma C, et al. Encoding word order in complex embeddings[J]. arXiv preprint arXiv:1912.12333, 2019.\n", "limitations": "The authors conclude that there are limitations that they cannot explain why quantum-inspired algorithms are better than classical algorithms. In addition, I think the limitation of this work is also the proposal of the regularization term, and whether the regularization term can be repurposed and better explained by quantum theory.\n", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "From the overall perspective, the idea of this paper is interesting, and the algorithm proposed is small and exquisite. The experimental results show that the training time is greatly shortened, and the experimental accuracy is improved compared with baseline models.\n\nHowever, the paper also has some weaknesses that need to be enhanced. First, the quantum-inspired models applied to supervised classification tasks is not the author's initiative. Moreover, in the experimental analysis, there is a lack of comparative experiments between this model and other quantum-inspired models. Second, the author uses quantum theory to analogize the relationship between documents and words, and the relationship between words and categories. This approach is based on intuition and has no strong theoretical basis. From the perspective of quantum measurement, this method is not reasonable. Third, the regularization lacks detailed analysis, and the use of entropy is incomprehensible. Fourth, some important information is not detailed in this paper, such as the necessity of using quantum theory, the method of quantum theory modeling prior knowledge, and what information the prior knowledge represents in the task. These makes people doubt whether it is necessary to introduce quantum theory into the problem raised in this paper. \n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "3uCarveSVBL", "reviewer": "Reviewer_mSUa", "summary": "This paper proposes a novel text classification method (Born layer) based on quantum mechanics. The Born layer takes the input vector as the wave function and predicts the class of the given input with Born’s rule. The Born layer can be applied to any other neural network or function that transforms the input text into a feature vector. This paper uses the simple text transformation method, tokenization, for the transformation function, with the square root function derived by the Born rule. Experiments on three text classification datasets (20Newsgroup, R8, and R52) show the efficacy of the proposed method. The proposed method is also computationally efficient.", "questions": "1. Line 67: Is x_j unnormalized probability? Eq 4 states that x_j is a probability.\n2. The Born layer is one of the main contributions of this paper. Do authors try any ablation study on the Born layer? e.g., apply the Born layer to other approaches and see the performance improvement.\n\nComments\n\n1. In line 79, this paper describes P as the probability, but in line 83, the authors describe P as a vector. Please provide a more clear description.\n2. Please provide a more description for the generalization form. What do the authors want to claim from the ablation study for the hyperparameters a and b?\n3. The proposed method is applicable to other text classification tasks such as sentiment classification. More experiments on other popular text classification problems are recommended.", "limitations": "The authors have addressed the limitations. No negative societal impact.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n1. The proposed method is novel.\n2. This paper is the first trial that applies quantum theory to the text classification problem.\n3. Experimental results support the validity of the proposed method.\n\nWeaknesses\n\n1. This paper is hard to follow. See the questions section for more details.\n2. This paper does not provide any detailed description of their assumption that “text is a superposition of words.” What is the motivation that the authors see a text snippet as a superposition of words? How is this assumption brings a positive impact on text classification? What problem do existing studies have, and how does the proposed method resolve it?", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept