Ontology-based n-ball Concept Embeddings Informing Few-shot Image Classi cation

Abstract

We propose a novel framework named ViOCE that integrates ontology-based background knowledge in the form of n-ball concept embeddings into a neural network based vision architecture. The approach consists of two components - converting symbolic knowledge of an ontology into continuous space by learning n-ball embeddings that capture properties of subsumption and disjointness, and guiding the training and inference of a vision model using the learnt embeddings. We evaluate ViOCE using the task of few-shot image classi cation, where it demonstrates superior performance on two standard benchmarks.

Keywords

• Background Knowledge
• Ontology
• Machine Learning
• Few-shot Learning

Introduction

Ontologies can capture consistent, generalised and structured knowledge that can be used with reasoning tools [ 23 ] that ensure knowledge consistency together with the ability to infer new knowledge [ 1 ]. Sometimes knowledge graphs are also called as ontologies [ 30 ], but we identify clear di erences. Knowledge graphs tend to be more loosely de ned, whereas ontologies have a well-de ned semantics that distinguish concepts from the given knowledge speci cation and other relationships (e.g., hasPart) between concepts bound by logical axioms. Sometimes a knowledge graph can be seen as a speci c instantiation of a whole or part of an ontology representing only object-level information [ 15 ], whereas ontologies include both concept-level information and objects or terms. Moreover with powerful reasoning tools, ontologies facilitate the discovery of implicit knowledge from explicitly de ne knowledge. This study sheds light on the use of ontologies in a machine learning context. We use the Web Ontology Language (OWL) [ 18 ] in constructing our ontologies in this study. In order to assess the impact of knowledge integration to a visual recognition task, we chose few-shot image classi cation [ 2 ] to be the main task in this study. Few-shot learning in

Fig. 1. The proposed approach classi es images by projecting them towards concept nballs de ned in high-dimensional space according to ontology-based background knowledge. (a) shows a snapshot of a few predictions for three miniImageNet[ 27 ] classes `miniature poodle', `hotdog' and `street sign' made by a model trained using the ViOCE framework during few-shot image classi cation. The dimensionality of the n-balls is reduced to 2 for visualisation purposes. A correct prediction is an image projected to be inside of the n-ball of its ground truth label. Additionally, the surrounding n-balls to the ground truth n-balls, de ned according to the background knowledge, gives us an unique opportunity to measure the `certainty' of the model in classifying each image. For example, not all `miniature poodle' images lie inside the ground truth n-ball but an image lying inside `dog' can be identi ed as semantically meaningful. (b) is a visualisation of the same set of n-balls in (a) reduced to a 3-dimensional space in order to provide a clearer idea on the nature of n-ball shape and placing. an image classi cation context focuses on e ectively learning the visual features of a class with very few examples.

The proposed ViOCE framework, we adopt a technique to embed ontologybased knowledge as n-balls inspired by the work done by Kulmanov et al. [ 14 ]. This embedding can represent specialisations (e.g., Dog SubclassOf Animal) using the property of one n-ball enclosing another and partonomies (e.g., Dog hasPart Tail) using translations of n-ball positions. In this study, we directly utilise two loss design components of [ 14 ] to capture subsumption and disjointness axioms, while extending their approach with more regularisation components in order to embed large hierarchies in a favourable manner for a downstream vision task. Additionally, we propose the use of the inferred class hierarchy of the input ontology and introduce a technique to evaluate the quality of the learnt embeddings during the embedding learning process. The learnt n-ball embeddings can be seen as de nitions of space for each concept in consideration that preserves the inferred class hierarchy entailed by the ontology. Next, we introduce a method to use a vision model [5; 12] to map input images to the space de ned by the concept embeddings, informing the vision task with the knowledge captured from the ontology. Figure 1 shows a snapshot of a few predictions for some miniImageNet[ 27 ] classes `miniature poodle', `hotdog' and `street sign' made by a model trained using the ViOCE framework. We nd that our approach facilitates better transparency on the behaviour of both knowledge embeddings and visual feature learning.

Overall, we extend [ 14 ] to capture knowledge from an ontology in the form of n-ball embeddings and show that they are favourable for the downstream vision task of few-shot image classi cation. This is also coupled with a technique to measure the quality of the learnt embeddings with respect to the knowledge entailed by the ontology. Next, we propose a technique to utilise the n-balls to guide a vision model during its training and inference stages performing few-shot image classi cation. 2

Related Work

An area that inspires the investigation of background knowledge integration in vision is the existing work done in knowledge-based vision systems [13; 25]. In [ 13 ], an interesting categorisation of knowledge that can be used as background knowledge is proposed, namely, permanent theoretical knowledge, circumstantial knowledge, subjective experimental knowledge and data knowledge. Although how these categories are formed is debatable, the importance of looking into di erent forms of knowledge that can be used as background knowledge is identi ed. The choice of knowledge form can be very much based on the considered vision application, as pointed out in [ 25 ], where the authors curate a number of vision tasks along with the forms of knowledge used to inform the learning process. Out of these, the use of scene graphs, probabilistic ontologies and rstorder logic rules grab the attention as promising paths to explore. Investigations into the use of background knowledge in the form of rst-Order Logic (FOL) is prominently seen in several studies [ 10 ]. As shown in [ 10 ], adaptation of logical knowledge as constraints during the learning process has generated promising results, that reinforces the attempts to use ontologies as background knowledge. The area of neuro-symbolic approaches also provides insights into the use of logical knowledge during the training of arti cial neural networks [ 22 ].

In terms of combining other sources of knowledge [ 16 ] with computer vision, this study is motivated by work such as [5; 12] and [ 29 ], where image features are mapped to a vector space de ned by language embeddings. This is identi ed as informing the image model with more knowledge that do not exist merely in the image features. In the case of [ 5 ], the knowledge from an unstructured text corpus is captured in the form of word embeddings to be integrated to the vision architecture. These approaches were mostly evaluated on zero-shot image classi cation, making use of the distance between points in the vector space de ned. These ndings motivate the proposed approach in this study, since they allow to extend standard vision models to incorporate language information. In terms of evaluation however, it can be argued that few-shot image classi cation [ 20 ] is a better candidate to measure how additional knowledge could help grasp new concepts faster. In terms of few-shot learning [4; 11], our study is motivated by metric learning methods [28; 20] due of their ability to extend standard vision architectures [ 6 ]. These approaches exploits image feature similarities [ 24 ] when learning and predicting a vision task. 3

n-Balls and EL Embeddings The mathematical concept of ball refers to the volume space bounded by a sphere and is also called a solid sphere. An n-ball usually refers to a ball in an ndimensional Euclidean space. The EL embeddings study [ 14 ] attempts to encode logical axioms by positioning n-balls. We explain how it works for encoding subsumption and disjointness as they are the most relevant to our work. Each concept P is embedded as an n-ball with its centre denoted by cP 2 Rn and the radius by rP 2 R. The basic idea is to move one ball inside the other for subsumption and to push two balls to stay away for disjointness. The following loss is minimized to encode O j= P v Q: lP vQ(cP ; cQ; rP ; rQ) = max(0; kcP + kcP k2 where k k2 denotes the l2 norm and 2 R is a user-set hyperparameter. It enforces the inequality kcP cQk2 rQ rP + , meanwhile regulates the ball centers to be close to a unit sphere. Through controlling the sign of , the user can adjust whether to push the P ball completely inside the Q ball. In a similar fashion, the loss for encoding O j= P u Q v ? is given as lP uQv?(cP ; cQ; rP ; rQ) = max(0; kcP It enforces the inequality kcP cQk2 rQ + rP + . According to the setting of , the user can decide how far the two balls are pushed away. 4

Proposed Method: ViOCE

We study how to e ectively integrate ontology-based background knowledge to improve few-shot image classi cation. More speci cally, this paper is focused on using additional hierarchical knowledge about the di erent classes to help image classi cation, achieving reduced data dependency of vision model architectures that are based on deep neural networks.

Adopting few-shot image classi cation as our benchmark [ 9 ], we train a neural vision model using a set of background images BI = f(Ii; yi)gim=1 (base set) from K classes with yi 2 CB = fc1; c2; : : : cKg and a set of few-shot images (1) (2) F I = f(Ii; yi)gis=1 (novel set) from w classes with yi 2 CF = fc~1; c~2; : : : ; c~wg, where CB \ CF = ;, and Ii denotes the raw image vectors containing pixel values. The few-shot success is usually assessed by how accurate a model can select a correct class from the candidate class set CF for a new image from the few-shot classes. This is often referred to as the w-way s-shot few-shot image classi cation. We construct an ontology O by using the class label information CB and CF , and also WordNet. It provides information on relationships that can exist among the class labels, containing knowledge regarding to \SubClassOf" and \DisjointClasses". These de ne the subsumption and disjointness axioms in the ontology.

We propose ViOCE as a framework to improve few-shot image classi cation by integrating information provided by O, BI and F I. It is composed of two main components: (1) to embed classes in CB and CF as n-balls based on the constructed O, (2) to embed images in the same Euclidean space as the n-balls with a suitable arrangement, and to infer the class for a query image based on its image embedding and the n-ball embeddings of the candidate classes. Figure 2 shows the general framework ow with an overview of all processes and data inputs. 4.1

Concept n-Ball Embeddings

We build upon the EL embedding technique [ 14 ] to learn a set of n-balls for all concepts Oe in the ontology O, which is referred to as a concept embedding. We extract subsumption and disjointness axioms to de ne the class hierarchy of the ontology O. It has been noticed that the entailed transitive relations such as if Poodle SubclassOf Dog and Dog SubclassOf Animal, then Poodle SubclassOf Animal are usually not well re ected by the learned n-balls. To overcome this, we use the inferred class hierarchy (ICH). Assuming all the concepts are satis able with O, the ICH is computed according to Equation 3. ICH contains all possible subsumption relations according to the de nition of O.

ICH(O) = fP v QjP 6= Q; P; Q 2 Oe; O j= P v Qg: (3)

If simply to follow Eqs. (1) and (2), the radius of the learned n-ball for a leaf concept, which corresponds to an image class in CB or CF , can end up being very small, in order to t into the balls of its ancestor concepts. Since in the image embedding learning, we will map each image as a data point inside the n-ball corresponding to its ground truth class, an overly small radius can a ect the learning accuracy. To tackle this, we introduce a regularisation term in Eq. (5) to prevent radius shrinkage. Also, the embedding quality can deteriorate as the class hierarchy of the input ontology becomes larger. To improve the embedding quality, we introduce an extra hyperparameter in Eq. (4) to explore potentially more expressive design spaces, which is supported by an additional parameter (4) (5) (6) tuning process. Finally, we minimise the following loss function: = lc fcP gP 2Oe; frP gP 2Oe

X max(0; kcP Here, Nh denotes the total level number contained by the class hierarchy, and L(P ) denotes the level of the concept P in the hierarchy, e.g.,, the top-most concept has level 1. N (P ) denotes the number of times the concept P appears in the extracted axioms. Both ; > 0 are hyperparameters. Eq. (5) restrict the radius of the concept P 's n-ball to be no less than pNh L(P ). The top-level concepts are allowed to have larger n-balls than the bottom ones. 4.2

Hyperparameter Tuning of n-ball Embeddings

Three parameter tuning scores are proposed by examining whether kcP cQk rQ rP holds for a ground truth subsumption ICH(O) j= P v Q. All the ground truth subsumptions are considered as positive instances. If the inequality holds, it is considered as a positive prediction. The classical F1 score, which is the harmonic mean of the precision and recall, is used to assess the prediction accuracy of these subsumptions. We calculate two versions of F1 score, one is referred to as F (all) based on all the subsumptions extracted from ICH(O).

1 The other only considers the subsumptions involving the leaf concepts, which correspond to all the classes in CB and CF , as well as their direct parent classes. This score is referred to as F1(leaf). The third parameter tuning score SD examines the disjointness between the leaf concepts. Enumerating all the pairs of leaf concepts, SD is equal to the number of pairs for which the condition kcP cQk rP + rQ holds. A higher SD indicates less overlapping between the n-balls of the leaf concepts. We need SD to be greater that a threshold value of T .

A good concept embedding result should have high F (all), F (leaf) and SD 1 1 scores. We compute these scores as a mandatory step at the end of each embedding learning process. The hyperparameters governing the scores are , and . We use grid search to nd the best combination of these parameters that would result in the best F (all), F (leaf) and SD scores.

1 1

4.3 Image Embedding Learning

Our vision model is composed of a base DCNN architecture coupled with a multi-layer perceptron (MLP). The DCNN computes the visual features for an image by taking its raw pixel representation vector as the input: fi = D(Ii; D) where fi 2 Rd. The MLP is responsible for mapping the visual features fi to the n-dimensional Euclidean space where the n-ball concept embeddings sit: hi = M(fi; M) where hi 2 Rn. We use D and M to denote the neural network parameters to be trained for the DCNN and MLP, respectively. The idea is to identify visual features of an image (using a DCNN) so that they can be mapped (by an MLP) as a data point inside the n-ball of its ground truth class. For example, an image containing the visual features of a \poodle" should be mapped inside the n-ball of the \poodle" concept learnt from the ontology.

To achieve this, the following a pairwise ranking loss is used to optimise the network parameters: lI( D; M) = Pim=1 max (0; kcP hik2

rP ) + PQ2Ci( ) max(0; rQ kcQ hik2) ; (7) where ; > 0 are hyperparameters. The set Ci( ) contains the negative classes de ned for each image Ii of the positive class with its embedding computed by hi = M( D(Ii; D); M). When setting = = 1, the loss enforces kcP hik2 rP , pushing the embedded image point to stay inside the n-ball of the correct concept class P , while kcQ hik2 rQ, to stay outside the n-ball of the incorrect concept class Q. The hyperparameters and are placed to control the intensity of this e ect, e.g.,, < 1 requiring to lie closer to the center which makes the task harder.

A speci cation crucial to learning performance is the selection of negatives concepts in Ci( ). Following the notion of \hard negatives" in [ 19 ], we select \hard negatives" for each positive concepts based on similarity. For example, the \poodle" concept is more similar to \golden retriever" in contrast to the \street sign", therefore it is more challenging to distinguish between \poodle" and \golden retriever". So we choose as the hard negatives the more similar concepts to a positive concept. Speci cally, we evaluate similarities between concepts by Euclidean distances between the centre vectors of their corresponding n-balls, and perform k-means clustering based on these. After clustering the centre vectors of the leaf concepts (image classes), for each image class, all the other image classes from the same cluster as it are treated as the \hard negatives" and are included to Ci( ). In practice, we rst train the DCNN and MLP from scratch by minimising Eq. (7) using the background images BI. This is called base learning (BL). Then, we ne tune the MLP by using the few-shot images F I by minimising the same loss, but keep the weights of DCNN xed. This called the few-shot learning (FSL).

We test the vision model using the testing images of F I (F Ite) after the ne-tuning of MLP in the F SL stage. During inference, a prediction is made by nding the n-ball which an image feature projection lies in. Let U = kcP hk rP , where h is an output feature for a query image from the vision model and cP and rP are the centre and radius of a selected n-ball of P respectively. If U 0, we nd that h lies inside the n-ball of P . Hence the classi cation of h will be class P . In case some h does not lie inside any of the n-balls of the w classes in the few-shot task, we choose the closest lying n-ball centre ci out of the classes to h, where argminci(i=1;2;::;w) kci hk , as the prediction. The proportion of the correct predictions out of all images in F Ite is recorded as the accuracy of the vision model in this study. 5

Experiment Setting

MiniImageNet dataset consists of 60,000 images of 100 classes from ImageNet where each class carries 600 example images [ 27 ]. Following the same splitting as in [ 8 ], 80 and 20 classes were allocated for training and testing respectively. TieredImageNet dataset is larger in size than miniImageNet, containing 608 classes from ImageNet [ 21 ]. Its classes are acquired based on 34 higher-level categories. We use a training set consisting of 26 higher-level categories with 448 classes, and testing set of 8 higher-level categories with 160 classes.

We construct two new ontologies based on the image labels of the datasets for each few-shot image classi cation benchmark. All selected datasets are subsets of ImageNet [ 3 ], where WordNet [ 17 ] synsets are used to annotate all images. This o ered the opportunity to use the information from WordNet to formulate more knowledge about the image labels. We chose the hypernym tree of WordNet to be the source of the class hierarchy in this study, where given a label, the corresponding synset name together with all other synsets above it until the root (entity.n.01 ) was extracted. All these concepts were included in the ontology1. The dimensionality of the concept embeddings was chosen to be 300. During all experiments, ResNet50 [ 7 ] architecture was chosen to be the base network and the MLP was composed of 5 layers with sizes of 2048, 1024, 512, 512 and 300. 6

Results

6.1

Few-shot image classi cation results

ViOCE is evaluated by comparing with the performance of several existing approaches according to [ 26 ] under the same con guration. We conduct experiments for w = f5; 20g and s = f1; 5g. Table 1 reports the 5-way 1-shot and 5-shot performance comparisons. It can be seen that ViOCE surpasses the the performance of all other approaches in every 5-way tasks with both datasets, while achieving >90% accuracy in miniImageNet 5-shot task.

The study further extends the evaluation with the miniImageNet dataset to the task of 20-way 1-shot and 5-shot classi cation. In this case, considering all the 20 few-shot classes o ers a bigger challenge to the model, having to distinguish between more classes with a few examples. Table 2 presents the result comparison on this task. ViOCE surpasses the performance of existing approaches in both 1-shot and 5-shot tasks with comfortable margins.

Another interesting observation during the BL stage of ViOCE was the behaviour of the training and testing accuracies of the vision model. With miniImageNet for example, the model was trained with 500 images per class across 80 classes, which is comparable to a standard image classi cation task. The training and testing accuracies were 85.32% and 95.36% respectively. The higher testing accuracy demonstrates the better generalisation ability of the learnt model. We argue that this e ect is due to not forcing the image features to a xed point as done in a standard training setting. The n-ball embeddings de ne a volume of space for each class providing more exibility for the arrangement of image feature points. 7

Conclusion

We show that the introduction of ontology-based background knowledge to a visual model can improve its performance in the task of few-shot image classi cation. The proposed ViOCE framework is capable of utilising the n-ball concept embeddings in an e ective way to inform the training and inference procedures of a vision model, and producing superior performance on two benchmarks. In future, we plan to extend this study to evaluate the semantically meaningful errors in classi cation and utilise multi-relational knowledge when learning concept embeddings.

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