Bilinear models (Tenenbaum & Freeman, 2000)
provide richer representations than linear models. To exploit this advantage, fully-connected layers in neural networks can be replaced with bilinear pooling. The outer product of two vectors (or Kroneker product for matrices) is involved in bilinear pooling, as a result of this, all pairwise interactions among given features are considered. Recently, a successful application of this technique is used for fine-grained visual recognition(Lin et al., 2015).
However, bilinear pooling produces a high-dimensional feature of quadratic expansion, which may constrain a model structure and computational resources. For example, an outer product of two feature vectors, both of which have 1K-dimensionality, produces a million-dimensional feature vector. Therefore, for classification problems, the choice of the number of target classes is severely constrained, because the number of parameters for a standard linear classifier is determined by multiplication of the size of the high-dimensional feature vector and the number of target classes.
Compact bilinear pooling (Gao et al., 2016)
reduces the quadratic expansion of dimensionality by two orders of magnitude, retaining the performance of the full bilinear pooling. This approximation uses sampling-based computation, Tensor Sketch Projection(Charikar et al., 2002; Pham & Pagh, 2013), which utilizes an useful property that , which means the projection of outer product of two vectors is the convolution of two projected vectors. Here, is the proposed projection function, and, and are randomly sampled parameters by the algorithm.
Nevertheless, compact bilinear pooling embraces two shortcomings. One comes from the sampling approach. Compact bilinear pooling relies on a favorable property, , which provides a basis to use projected features instead of original features. Yet, calculating the exact expectation is computationally intractable, so, the random parameters, and are fixed during training and evaluation. This practical choice leads to the second. The projected dimension of compact bilinear pooling should be large enough to minimize the bias from the fixed parameters. Practical choices are 10K and 16K for 512 and 4096-dimensional inputs, respectively (Gao et al., 2016; Fukui et al., 2016). Though, these compacted dimensions are reduced ones by two orders of magnitude compared with full bilinear pooling, such high-dimensional features could be a bottleneck for computationally complex models.
We propose low-rank bilinear pooling using Hadamard product (element-wise multiplication), which is commonly used in various scientific computing frameworks as one of tensor operations. The proposed method factors a three-dimensional weight tensor for bilinear pooling into three two-dimensional weight matrices, which enforces the rank of the weight tensor to be low-rank. As a result, two input feature vectors linearly projected by two weight matrices, respectively, are computed by Hadamard product, then, followed by a linear projection using the third weight matrix. For example, the projected vector is represented by , where denotes Hadamard product.
We also explore to add non-linearity using non-linear activation functions into the low-rank bilinear pooling, and shortcut connections inspired by deep residual learning(He et al., 2016). Then, we show that it becomes a simple baseline model (Antol et al., 2015) or one-learning block of Multimodal Residual Networks (Kim et al., 2016b) as a low-rank bilinear model, yet, this interpretation has not be done.
Our contributions are as follows: First, we propose low-rank bilinear pooling to approximate full bilinear pooling to substitute compact bilinear pooling. Second, Multimodal Low-rank Bilinear Attention Networks (MLB) having an efficient attention mechanism using low-rank bilinear pooling is proposed for visual question-answering tasks. MLB achieves a new state-of-the-art performance, and has a better parsimonious property. Finally, ablation studies to explore alternative choices, e.g. network depth, non-linear functions, and shortcut connections, are conducted.
2 Low-rank Bilinear Model
Bilinear models use a quadratic expansion of linear transformation considering every pair of features.
where and are input vectors, is a weight matrix for the output , and is a bias for the output . Notice that the number of parameters is
including a bias vector, where is the number of output features.
Pirsiavash et al. (2009) suggest a low-rank bilinear method to reduce the rank of the weight matrix to have less number of parameters for regularization. They rewrite the weight matrix as where and , which imposes a restriction on the rank of to be at most .
Based on this idea, can be rewritten as follows:
where denotes a column vector of ones, and denotes Hadamard product. Still, we need two third-order tensors, and , for a feature vector , whose elements are . To reduce the order of the weight tensors by one, we replace with and with , then, redefine as and to get a projected feature vector . Then, we get:
are hyperparameters to decide the dimension of joint embeddings and the output dimension of low-rank bilinear models, respectively.
3 Low-rank Bilinear Pooling
A low-rank bilinear model in Equation 3 can be implemented using two linear mappings without biases for embedding two input vectors, Hadamard product to learn joint representations in a multiplicative way, and a linear mapping with a bias to project the joint representations into an output vector for a given output dimension. Then, we use this structure as a pooling method for deep neural networks. Now, we discuss possible variations of low-rank bilinear pooling based on this model inspired by studies of neural networks.
3.1 Full Model
3.2 Nonlinear Activation
Applying non-linear activation functions may help to increase representative capacity of model. The first candidate is to apply non-linear activation functions right after linear mappings for input vectors.
where denotes an arbitrary non-linear activation function, which maps any real values into a finite interval, e.g. sigmoid or . If two inputs come from different modalities, statistics of two inputs may be quite different from each other, which may result an interference. Since the gradient with respect to each input is directly dependent on the other input in Hadamard product of two inputs.
Additional applying an activation function after the Hadamard product is not appropriate, since activation functions doubly appear in calculating gradients. However, applying the activation function only after the Hadamard product would be alternative choice (We explore this option in Section 5) as follows:
Note that using the activation function in low-rank bilinear pooling can be found in an implementation of simple baseline for the VQA dataset (Antol et al., 2015) without an interpretation of low-rank bilinear pooling. However, notably, Wu et al. (2016c) studied learning behavior of multiplicative integration in RNNs with discussions and empirical evidences.
3.3 Shortcut Connection
When we apply two previous techniques, full model and non-linear activation, linear models of two inputs are nested by the non-linear activation functions. To avoid this unfortunate situation, we add shortcut connections as explored in residual learning (He et al., 2016).
where and are shortcut mappings. For linear projection, the shortcut mappings are linear mappings. Notice that this formulation is a generalized form of the one-block layered MRN (Kim et al., 2016b). Though, the shortcut connections are not used in our proposed model, as explained in Section 6.
4 Multimodal Low-rank Bilinear Attention Networks
In this section, we apply low-rank bilinear pooling to propose an efficient attention mechanism for visual question-answering tasks, based on the interpretation of previous section. We assumed that inputs are a question embedding vector and a set of visual feature vectors over lattice space.
4.1 Low-rank Bilinear Pooling in Attention Mechanism
Attention mechanism uses an attention probability distributionover lattice space. Here, using low-rank bilinear pooling, is defined as
where , , is a hyperbolic tangent function, , , , , and . If , multiple glimpses are explicitly expressed as in Fukui et al. (2016), conceptually similar to Jaderberg et al. (2015). And, the softmax function applies to each row vector of . The bias terms are omitted for simplicity.
4.2 Multimodal Low-rank Bilinear Attention Networks
Attended visual feature is a linear combination of with coefficients . Each attention probability distribution is for a glimpse . For , is the concatenation of resulting vectors as
denotes concatenation of vectors. The posterior probability distribution is an output of asoftmax function, whose input is the result of another low-rank bilinear pooling of and as
where denotes a predicted answer, is a set of candidate answers and is an aggregation of entire model parameters.
|* Sampled Answer||57.7M||64.80||83.59||38.38||54.73|
|* No Shortcut||51.9M||65.08||84.14||38.21||54.87|
|MCB+Att (Fukui et al., 2016)||69.2M||64.2||82.2||37.7||54.8|
|MCB+Att+GloVe (Fukui et al., 2016)||70.5M||64.7||82.5||37.6||55.6|
|MCB+Att+Glove+VG (Fukui et al., 2016)||70.5M||65.4||82.3||37.2||57.4|
In this section, we conduct six experiments to select the proposed model, Multimodal Low-rank Bilinear Attention Networks (MLB). Each experiment controls other factors except one factor to assess the effect on accuracies. Based on MRN (Kim et al., 2016b), we start our assessments with an initial option of and shortcut connections of MRN, called as Multimodal Attention Residual Networks (MARN). Notice that we use one embeddings for each visual feature for better performance, based on our preliminary experiment (not shown). We attribute this choice to the attention mechanism for visual features, which provides more capacity to learn visual features. We use the same hyper-parameters of MRN (Kim et al., 2016b), without any explicit mention of this.
The VQA dataset (Antol et al., 2015) is used as a primary dataset, and, for data augmentation, question-answering annotations of Visual Genome (Krishna et al., 2016) are used. Validation is performed on the VQA test-dev split, and model comparison is based on the results of the VQA test-standard split. For the comprehensive reviews of VQA tasks, please refer to Wu et al. (2016a) and Kafle & Kanan (2016a). The details about preprocessing, question and vision embedding, and hyperparameters used in our experiments are described in Appendix A. The source code for the experiments is available in Github repository111https://github.com/jnhwkim/MulLowBiVQA.
Number of Learning Blocks
Kim et al. (2016b) argue that three-block layered MRN shows the best performance among one to four-block layered models, taking advantage of residual learning. However, we speculate that an introduction of attention mechanism makes deep networks hard to optimize. Therefore, we explore the number of learning blocks of MARN, which have an attention mechanism using low-rank bilinear pooling.
Number of Glimpses
Fukui et al. (2016) show that the attention mechanism of two glimpses was an optimal choice. In a similar way, we assess one, two, and four-glimpse models.
VQA (Antol et al., 2015) dataset has ten answers from unique persons for each question, while Visual Genome (Krishna et al., 2016) dataset has a single answer for each question. Since difficult or ambiguous questions may have divided answers, the probabilistic sampling from the distribution of answers can be utilized to optimize for the multiple answers. An instance 222https://github.com/akirafukui/vqa-mcb/blob/5fea8/train/multi_att_2_glove/vqa_data_provider_layer.py#L130 can be found in Fukui et al. (2016). We simplify the procedure as follows:
where denotes the number of unique answer in a set of multiple answers, denotes a mode, which is the most frequent answer, and
denotes the secondly most frequent answer. We define the divided answers as having at least three answers which are the secondly frequent one, for the evaluation metric of VQA(Antol et al., 2015),
The rate of the divided answers is approximately , and only of questions have more than two divided answers in VQA dataset. We assume that it eases the difficulty of convergence without severe degradation of performance.
The contribution of shortcut connections for residual learning is explored based on the observation of the competitive performance of single-block layered model. Since the usefulness of shortcut connections is linked to the network depth (He et al., 2016).
The data augmentation with Visual Genome (Krishna et al., 2016) question answer annotations is explored. Visual Genome (Krishna et al., 2016) originally provides 1.7 Million visual question answer annotations. After aligning to VQA, the valid number of question-answering pairs for training is 837,298, which is for distinct 99,280 images.
The six experiments are conducted sequentially. Each experiment determines experimental variables one by one. Refer to Table 1, which has six sectors divided by mid-rules.
6.1 Six Experiment Results
Number of Learning Blocks
Though, MRN (Kim et al., 2016b) has the three-block layered architecture, MARN shows the best performance with two-block layered models (63.92%). For the multiple glimpse models in the next experiment, we choose one-block layered model for its simplicity to extend, and competitive performance (63.79%).
|iBOWIMG (Zhou et al., 2015)||55.89||76.76||34.98||42.62||61.97|
|DPPnet (Noh et al., 2016)||57.36||80.28||36.92||42.24||62.69|
|Deeper LSTM+Normalized CNN (Antol et al., 2015)||58.16||80.56||36.53||43.73||63.09|
|SMem (Xu & Saenko, 2016)||58.24||80.80||37.53||43.48||-|
Ask Your Neurons(Malinowski et al., 2016)
|SAN (Yang et al., 2016)||58.85||79.11||36.41||46.42||-|
|D-NMN (Andreas et al., 2016)||59.44||80.98||37.48||45.81||-|
|ACK (Wu et al., 2016b)||59.44||81.07||37.12||45.83||-|
|FDA (Ilievski et al., 2016)||59.54||81.34||35.67||46.10||64.18|
|HYBRID (Kafle & Kanan, 2016b)||60.06||80.34||37.82||47.56||-|
|DMN+ (Xiong et al., 2016)||60.36||80.43||36.82||48.33||-|
|MRN (Kim et al., 2016b)||61.84||82.39||38.23||49.41||66.33|
|HieCoAtt (Lu et al., 2016)||62.06||79.95||38.22||51.95||66.07|
|RAU (Noh & Han, 2016)||63.2||81.7||38.2||52.8||67.3|
Number of Glimpses
Compared with the results of Fukui et al. (2016), four-glimpse MARN (64.61%) is better than other comparative models. However, for a parsimonious choice, two-glimpse MARN (64.53%) is chosen for later experiments. We speculate that multiple glimpses are one of key factors for the competitive performance of MCB (Fukui et al., 2016), based on a large margin in accuracy, compared with one-glimpse MARN (63.79%).
The results confirm that activation functions are useful to improve performances. Surprisingly, there is no empirical difference between two options, before-Hadamard product and after-Hadamard product. This result may build a bridge to relate with studies on multiplicative integration with recurrent neural networks(Wu et al., 2016c).
Sampled answers (64.80%) result better performance than mode answers (64.53%). It confirms that the distribution of answers from annotators can be used to improve the performance. However, the number of multiple answers is usually limited due to the cost of data collection.
Though, MRN (Kim et al., 2016b) effectively uses shortcut connections to improve model performance, one-block layered MARN shows better performance without the shortcut connection. In other words, the residual learning is not used in our proposed model, MLB. It seems that there is a trade-off between introducing attention mechanism and residual learning. We leave a careful study on this trade-off for future work.
Data augmentation using Visual Genome (Krishna et al., 2016) question answer annotations significantly improves the performance by 0.76% in accuracy for VQA test-dev split. Especially, the accuracy of others (ETC)-type answers is notably improved from the data augmentation.
6.2 Comparison with State-of-the-Art
The comparison with other single models on VQA test-standard is shown in Table 2. The overall accuracy of our model is approximately 1.9% above the next best model (Noh & Han, 2016) on the Open-Ended task of VQA. The major improvements are from yes-or-no (Y/N) and others (ETC)-type answers. In Table 3, we also report the accuracy of our ensemble model to compare with other ensemble models on VQA test-standard, which won 1st to 5th places in VQA Challenge 2016333http://visualqa.org/challenge.html. We beat the previous state-of-the-art with a margin of 0.42%.
|RAU (Noh & Han, 2016)||64.12||83.33||38.02||53.37||67.34|
|MRN (Kim et al., 2016b)||63.18||83.16||39.14||51.33||67.54|
|DLAIT (not published)||64.83||83.23||40.80||54.32||68.30|
|Naver Labs (not published)||64.79||83.31||38.70||54.79||69.26|
|MCB (Fukui et al., 2016)||66.47||83.24||39.47||58.00||70.10|
|Human (Antol et al., 2015)||83.30||95.77||83.39||72.67||91.54|
7 Related Works
MRN (Kim et al., 2016b)
proposes multimodal residual learning with Hadamard product of low-rank bilinear pooling. However, their utilization of low-rank bilinear pooling is limited to joint residual mapping function for multimodal residual learning. Higher-order Boltzmann Machines(Memisevic & Hinton, 2007, 2010)
use Hadamard product to capture the interactions of input, output, and hidden representations for energy function.Wu et al. (2016c) propose the recurrent neural networks using Hadamard product to integrate multiplicative interactions among hidden representations in the model. For details of these related works, please refer to Appendix D.
7.1 Compact Bilinear Pooling
where denotes outer product, denotes convolution, , FFT
denotes Fast Fourier Transform,denotes an output dimension, , and are inputs, and and
are random variables.is sampled from , and is sampled from , then, both random variables are fixed for further usage. Even if the dimensions of and are different from each other, it can be used for multimodal learning (Fukui et al., 2016).
Similarly to Equation 1, compact bilinear pooling can be described as follows:
where if is sampled from , is sampled from , and the compact bilinear pooling is followed by a fully connected layer . Then, this method can be formulated as a hashing trick (Weinberger et al., 2009; Chen et al., 2015) to share randomly chosen bilinear weights using parameters for a output value, in a way that a single parameter is shared by
bilinear terms in expectation, with the variance of(See Appendix B).
In comparison with our method, their method approximates a three-dimensional weight tensor in bilinear pooling with a two-dimensional matrix , which is larger than the concatenation of three two-dimensional matrices for low-rank bilinear pooling. The ratio of the number of parameters for a single output to the total number of parameters for outputs is (Fukui et al., 2016), vs. (ours), since our method uses a three-way factorization. Hence, more parameters are allocated to each bilinear approximation than compact bilinear pooling does, effectively managing overall parameters guided by back-propagation algorithm.
MCB (Fukui et al., 2016), which uses compact bilinear pooling for multimodal tasks, needs to set the dimension of output to 16K, to reduce the bias induced by the fixed random variables and . As a result, the majority of model parameters (16K 3K = 48M) are concentrated on the last fully connected layer, which makes a fan-out structure. So, the total number of parameters of MCB is highly sensitive to the number of classes, which is approximately 69.2M for MCB+att, and 70.5M for MCB+att+GloVe. Yet, the total number of parameters of our proposed model (MLB) is 51.9M, which is more robust to the number of classes having = 1.2K, which has a similar role in model architecture.
We suggest a low-rank bilinear pooling method to replace compact bilinear pooling, which has a fan-out structure, and needs complex computations. Low-rank bilinear pooling has a flexible structure using linear mapping and Hadamard product, and a better parsimonious property, compared with compact bilinear pooling. We achieve new state-of-the-art results on the VQA dataset using a similar architecture of Fukui et al. (2016), replacing compact bilinear pooling with low-rank bilinear pooling. We believe our method could be applicable to other bilinear learning tasks.
The authors would like to thank Patrick Emaase for helpful comments and editing. Also, we are thankful to anonymous reviewers who provided comments to improve this paper. This work was supported by NAVER LABS Corp. & NAVER Corp. and partly by the Korea government (IITP-R0126-16-1072-SW.StarLab, KEIT-10044009-HRI.MESSI, KEIT-10060086-RISF, ADD-UD130070ID-BMRR). The part of computing resources used in this study was generously shared by Standigm Inc.
- Andreas et al. (2016) Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Learning to Compose Neural Networks for Question Answering. arXiv preprint arXiv:1601.01705, 2016.
Antol et al. (2015)
Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra,
C. Lawrence Zitnick, and Devi Parikh.
VQA: Visual Question Answering.
IEEE International Conference on Computer Vision, 2015.
- Charikar et al. (2002) Moses Charikar, Kevin Chen, and Martin Farach-Colton. Finding frequent items in data streams. In International Colloquium on Automata, Languages, and Programming, pp. 693–703. Springer, 2002.
Chen et al. (2015)
Wenlin Chen, James T. Wilson, Stephen Tyree, Kilian Q. Weinberger, and Yixin
Compressing Neural Networks with the Hashing Trick.
32nd International Conference on Machine Learning, pp. 2285–2294, 2015.
Cho et al. (2014)
Kyunghyun Cho, Bart Van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau,
Fethi Bougares, Holger Schwenk, and Yoshua Bengio.
Learning Phrase Representations using RNN Encoder-Decoder for
Statistical Machine Translation.
2014 Conference on Empirical Methods in Natural Language Processing, pp. 1724–1734, 2014.
- Fukui et al. (2016) Akira Fukui, Dong Huk Park, Daylen Yang, Anna Rohrbach, Trevor Darrell, and Marcus Rohrbach. Multimodal Compact Bilinear Pooling for Visual Question Answering and Visual Grounding. arXiv preprint arXiv:1606.01847, 2016.
- Gal (2015) Yarin Gal. A Theoretically Grounded Application of Dropout in Recurrent Neural Networks. arXiv preprint arXiv:1512.05287, 2015.
Gao et al. (2016)
Yang Gao, Oscar Beijbom, Ning Zhang, and Trevor Darrell.
Compact Bilinear Pooling.
IEEE Conference on Computer Vision and Pattern Recognition, 2016.
- He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image Recognition. In IEEE Conference on Computer Vision and Pattern Recognition, 2016.
- Hochreiter & Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. Long Short-Term Memory. Neural computation, 9(8):1735–1780, 1997.
- Ilievski et al. (2016) Ilija Ilievski, Shuicheng Yan, and Jiashi Feng. A Focused Dynamic Attention Model for Visual Question Answering. arXiv preprint arXiv:1604.01485, 2016.
- Jaderberg et al. (2015) Max Jaderberg, Karen Simonyan, Andrew Zisserman, and Koray Kavukcuoglu. Spatial Transformer Networks. In Advances in Neural Information Processing Systems 28, pp. 2008–2016, 2015.
- Kafle & Kanan (2016a) Kushal Kafle and Christopher Kanan. Visual Question Answering: Datasets, Algorithms, and Future Challenges. arXiv preprint arXiv:1610.01465, 2016a.
- Kafle & Kanan (2016b) Kushal Kafle and Christopher Kanan. Answer-Type Prediction for Visual Question Answering. IEEE Conference on Computer Vision and Pattern Recognition, pp. 4976–4984, 2016b.
Kim et al. (2016a)
Jin-Hwa Kim, Jeonghee Kim, Jung-Woo Ha, and Byoung-Tak Zhang.
TrimZero: A Torch Recurrent Module for Efficient Natural Language Processing.In KIIS Spring Conference, volume 26, pp. 165–166, 2016a.
- Kim et al. (2016b) Jin-Hwa Kim, Sang-Woo Lee, Dong-Hyun Kwak, Min-Oh Heo, Jeonghee Kim, Jung-Woo Ha, and Byoung-Tak Zhang. Multimodal Residual Learning for Visual QA. arXiv preprint arXiv:1606.01455, 2016b.
- Kiros et al. (2015) Ryan Kiros, Yukun Zhu, Ruslan Salakhutdinov, Richard S. Zemel, Antonio Torralba, Raquel Urtasun, and Sanja Fidler. Skip-Thought Vectors. In Advances in Neural Information Processing Systems 28, pp. 3294–3302, 2015.
- Krishna et al. (2016) Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalantidis, Li-Jia Li, David A Shamma, Michael Bernstein, and Li Fei-Fei. Visual genome: Connecting language and vision using crowdsourced dense image annotations. arXiv preprint arXiv:1602.07332, 2016.
- Léonard et al. (2015) Nicholas Léonard, Sagar Waghmare, Yang Wang, and Jin-Hwa Kim. rnn : Recurrent Library for Torch. arXiv preprint arXiv:1511.07889, 2015.
- Lin et al. (2015) Tsung-Yu Lin, Aruni RoyChowdhury, and Subhransu Maji. Bilinear CNN Models for Fine-grained Visual Recognition. In IEEE International Conference on Computer Vision, pp. 1449–1457, 2015.
- Lu et al. (2016) Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Hierarchical Question-Image Co-Attention for Visual Question Answering. arXiv preprint arXiv:1606.00061, 2016.
- Malinowski et al. (2016) Mateusz Malinowski, Marcus Rohrbach, and Mario Fritz. Ask Your Neurons: A Deep Learning Approach to Visual Question Answering. arXiv preprint arXiv:1605.02697, 2016.
- Memisevic & Hinton (2007) Roland Memisevic and Geoffrey E Hinton. Unsupervised learning of image transformations. In IEEE Conference on Computer Vision and Pattern Recognition, 2007.
- Memisevic & Hinton (2010) Roland Memisevic and Geoffrey E Hinton. Learning to represent spatial transformations with factored higher-order Boltzmann machines. Neural computation, 22(6):1473–1492, 2010.
- Noh & Han (2016) Hyeonwoo Noh and Bohyung Han. Training Recurrent Answering Units with Joint Loss Minimization for VQA. arXiv preprint arXiv:1606.03647, 2016.
Noh et al. (2016)
Hyeonwoo Noh, Paul Hongsuck Seo, and Bohyung Han.
Image Question Answering using Convolutional Neural Network with Dynamic Parameter Prediction.In IEEE Conference on Computer Vision and Pattern Recognition, 2016.
- Pham & Pagh (2013) Ninh Pham and Rasmus Pagh. Fast and scalable polynomial kernels via explicit feature maps. In 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 239–247. ACM, 2013.
- Pirsiavash et al. (2009) Hamed Pirsiavash, Deva Ramanan, and Charless C. Fowlkes. Bilinear classifiers for visual recognition. In Advances in Neural Information Processing Systems 22, pp. 1482–1490, 2009.
- Tenenbaum & Freeman (2000) Joshua B Tenenbaum and William T Freeman. Separating style and content with bilinear models. Neural computation, 12(6):1247–1283, 2000.
Tieleman & Hinton (2012)
Tijmen Tieleman and Geoffrey Hinton.
Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude.COURSERA: Neural Networks for Machine Learning, 4, 2012.
- Weinberger et al. (2009) Kilian Weinberger, Anirban Dasgupta, John Langford, Alex Smola, and Josh Attenberg. Feature hashing for large scale multitask learning. In 26th International Conference on Machine Learning, pp. 1113–1120, 2009.
- Wu et al. (2016a) Qi Wu, Damien Teney, Peng Wang, Chunhua Shen, Anthony Dick, and Anton van den Hengel. Visual Question Answering: A Survey of Methods and Datasets. arXiv preprint arXiv:1607.05910, 2016a.
- Wu et al. (2016b) Qi Wu, Peng Wang, Chunhua Shen, Anthony Dick, and Anton van den Hengel. Ask Me Anything: Free-form Visual Question Answering Based on Knowledge from External Sources. In IEEE Conference on Computer Vision and Pattern Recognition, 2016b.
- Wu et al. (2016c) Yuhuai Wu, Saizheng Zhang, Ying Zhang, Yoshua Bengio, and Ruslan Salakhutdinov. On Multiplicative Integration with Recurrent Neural Networks. arXiv preprint arXiv:1606.06630, 2016c.
- Xiong et al. (2016) Caiming Xiong, Stephen Merity, and Richard Socher. Dynamic Memory Networks for Visual and Textual Question Answering. In 33rd International Conference on Machine Learning, 2016.
- Xu & Saenko (2016) Huijuan Xu and Kate Saenko. Ask, Attend and Answer: Exploring Question-Guided Spatial Attention for Visual Question Answering. In European Conference on Computer Vision, 2016.
- Yang et al. (2016) Zichao Yang, Xiaodong He, Jianfeng Gao, Li Deng, and Alex Smola. Stacked Attention Networks for Image Question Answering. In IEEE Conference on Computer Vision and Pattern Recognition, 2016.
- Zhou et al. (2015) Bolei Zhou, Yuandong Tian, Sainbayar Sukhbaatar, Arthur Szlam, and Rob Fergus. Simple Baseline for Visual Question Answering. arXiv preprint arXiv:1512.02167, 2015.
A Experiment Details
We follow the preprocessing procedure of Kim et al. (2016b). Here, we remark some details of it, and changes.
a.1.1 Question Embedding
The 90.45% of questions for the 2K-most frequent answers are used. The vocabulary size of questions is 15,031. GRU (Cho et al., 2014) is used for question embedding. Based on earlier studies (Noh et al., 2016; Kim et al., 2016b), a word embedding matrix and a GRU are initialized with Skip-thought Vector pre-trained model (Kiros et al., 2015). As a result, question vectors have 2,400 dimensions.
a.2 Vision Embedding
The hyperparameters used in MLB of Table 2 are described in Table 4. The batch size is 100, and the number of iterations is fixed to 250K. For data augmented models, a simplified early stopping is used, starting from 250K to 350K-iteration for every 25K iterations (250K, 275K, 300K, 325K, and 350K; at most five points) to avoid exhaustive submissions to VQA test-dev evaluation server. RMSProp (Tieleman & Hinton, 2012) is used for optimization.
|14||attention lattice size|
|2,400||question embedding size|
|2,048||channel size of extracted visual features|
|1,200||joint embedding size|
|2||number of glimpses|
|2,000||number of candidate answers|
|0.99997592083||learning rate decay factor at every iteration|
|10||gradient clipping threshold|
a.4 Model Schema
Figure 1 shows a schematic diagram of MLB, where denotes Hadamard product, and denotes a linear combination of visual feature vectors using coefficients, which is the output of softmax function. If , the softmax function is applied to each row vectors of an output matrix (Equation 8), and we concatenate the resulting vectors of the linear combinations (Equation 9).
a.5 Ensemble of Seven Models
The test-dev results for individual models consisting of our ensemble model is presented in Table 6.
B Understanding of Multimodal Compact Bilinear Pooling
and are the given inputs, is the output. Random variables and are uniformly sampled from , and and are uniformly sampled from . Then, Count Sketch projection function (Charikar et al., 2002) projects and to intermediate representations and , which is defined as:
Notice that both and remain as constants after initialization (Fukui et al., 2016).
The probability of and for the given is . Hence, the expected number of bilinear terms in is . Since, the output is a result of circular convolution of and , the expected number of bilinear terms in is . Likewise, the probability of that a bilinear term is allocated in is . The probability distribution of the number of bilinear terms in follows a multinomial distribution, whose mean is and variance is .
Linear projection after the multimodal compact bilinear pooling provides weights on the bilinear terms, in a way that a shared weight is assigned to , which has bilinear terms in expectation, though each bilinear term can have a different sign induced by both and .
HashedNets (Chen et al., 2015) propose a method to compress neural networks using a low-cost hashing function (Weinberger et al., 2009), which is the same function of . They randomly group a portion of connections in neural networks to share a single weight. We speculate that multimodal compact bilinear pooling uses the hashing tick to reduce the number of full bilinear weights with the rate of . However, this approximation is limited to two-way interaction, compared with three-way factorization in our method.
C Replacement of Low-rank Bilinear Pooling
For the explicit comparison with compact bilinear pooling, we explicitly substitute compact bilinear pooling for low-rank bilinear pooling to control everything else, which means that the rest of the model architecture is exactly the same.
According to Fukui et al. (2016), we use MCB followed by Signed Square Root, L2-Normalization, Dropout (=0.1), and linear projection from 16,000-dimension to the target dimension. Also, Dropout (=0.3) for a question embedding vector. Note that an overall architecture for multimodal learning of both is the same. Experimental details are referenced from the implementation 444https://github.com/akirafukui/vqa-mcb of Fukui et al. (2016).
For test-dev split, our version of MCB gets 61.48% for overall accuracy (yes/no: 82.48%, number: 37.06%, and other: 49.07%) vs. 65.08% (ours, MLB in Table 1 ). Additionally, if the nonlinearity in getting attention distributions is increased as the original MCB does using ReLU, we get 62.11% for overall accuracy (yes/no: 82.55%, number: 37.18%, and other: 50.30%), which is still the below of our performance
). Additionally, if the nonlinearity in getting attention distributions is increased as the original MCB does using ReLU, we get 62.11% for overall accuracy (yes/no: 82.55%, number: 37.18%, and other: 50.30%), which is still the below of our performance555Our version of MCB definition can be found in https://github.com/jnhwkim/MulLowBiVQA/blob/master/netdef/MCB.lua.
We do not see it as a decisive evidence of the better performance of MLB, but as a reference (the comparison of test-dev results may be also unfair.), since an optimal architecture and hyperparameters may be required for each method.
D Related Works
d.1 Multimodal Residual Networks
where are parameter matrices, is the number of learning blocks, , , and . Notice that these equations can be generalized by Equation 7.
However, an explicit attention mechanism allows the use of lower-level visual features than fully-connected layers, and, more importantly, spatially selective learning. Recent state-of-the-art methods use a variant of an explicit attention mechanism in their models (Lu et al., 2016; Noh & Han, 2016; Fukui et al., 2016). Note that shortcut connections of MRN are not used in the proposed Multimodal Low-rank Bilinear (MLB) model. Since, it does not have any performance gain due to not stacking multiple layers in MLB. We leave the study of residual learning for MLB for future work, which may leverage the excellency of bilinear models as suggested in Wu et al. (2016a).
d.2 Higher-Order Boltzmann Machines
A similar model can be found in a study of Higher-Order Boltzmann Machines (Memisevic & Hinton, 2007, 2010). They suggest a factoring method for the three-way energy function to capture correlations among input, output, and hidden representations.
Setting aside of bias terms, the parameter tensor of unfactored Higher-Order Boltzmann Machines is replaced with three matrices, , , and .
d.3 Multiplicative Integration with Recurrent Neural Networks
Most of recurrent neural networks, including vanilla RNNs, Long Short Term Memory networks (Hochreiter & Schmidhuber, 1997)(Cho et al., 2014), share a common expression as follows:
where is a non-linear function, , , , , and is a bias vector. Note that, usually, is an input state vector and is an hidden state vector in recurrent neural networks.
Wu et al. (2016c) propose a new design to replace the additive expression with a multiplicative expression using Hadamard product as
Moreover, a general formulation of this multiplicative integration can be described as
which is reminiscent of full model in Section 3.1.