Improving Deep Learning with Lorentzian Geometry: Results from LHIER Experiments

Table of Links

Abstract and 1. Introduction

1. Related Work

2. Methodology

   3.1 Background

   3.2 Riemannian Optimization

   3.3 Towards Efficient Architectural Components

3. Experiments

   4.1 Hierarchical Metric Learning Problem

   4.2 Standard Classification Problem

4. Conclusion and References

4 Experiments

4.1 Hierarchical Metric Learning Problem

\ In the following experiment, we extend HIER to the Lorentz model (LHIER) and compare against the results provided by [20]. The aim of this is proving the effectiveness of our Lorentzian AdamW optimizer, general optimization scheme, curvature learning, and max distance scaling.

\ Experimental Setup We follow the experimental setup in Kim et al. [20] and rely on four main datasets: CUB-200- 2011 (CUB)[42], Cars-196 (Cars)[22], Stanford Online Product (SOP)[34], and In-shop Clothes Retrieval (InShop)[24]. Performance is measured using Recall@k which is the fraction of queries with one or more relevant samples in their k-nearest neighbors. Additionally, all model backbones are pretrained on Imagenet to ensure fair comparisons with the previous work.

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\ Moving to Lorentz Space We adapt the HIER to the hyperboloid using three major changes. We first replace the Euclidean linear layer with a Lorentzian linear layer in the model neck and implement our max distance scaling operation after. We then set the hierarchical proxies as learnable hyperbolic parameters and optimize them directly on the manifold using our Lorentzian AdamW. Finally, we change the Poincaré distance to the Lorentz distance for the LHIER loss and set the hierarchical proxies to be scaled beforehand. We also continue to use on fp16 precision which shows our model is more robust to stability and imprecision issues.

\ Results As show in table 1, our HIER+ model manages to improve performance in almost all scenarios. However, the percentage change varies depending on the dataset and the model used. While we do the best in the case of Resnet models, we are particularly worse for the DeiT model, especially at the higher dimensionality where our method is out performed in most datasets by HIER. This could be the issue of a lack of hyperparameter tuning, specifically in the case of the tanh scaling; the s factor which controls the tightness of the outputs. Kim et al. [20] control this through the use of norm clipping with varying intensities, a similar approach could be adopted to study the best-fitting scaling factors given the experimental settings.

4.2 Standard Classification Problem

Experimental Setup We follow the experimental setup of Bdeir et al. [1] and rely on three main datasets: CIFAR10, CIFAR100, Mini-Imagenet. We also extend the experiments to include Resnet50, however, due to the increased memory cost, we are only able to compare to the base Euclidean counterpart and hybrid models. For the Resnet-18 models we do no use our Lorentz core blocks and instead recreate the hybrid encoder similar to Bdeir et al. [1]. For all models, we use the efficient convolutional layer, all hyperbolic vectors are scaled using our max distance rescaling with s = 2. Additionally, curvature learning is performed for both our Resnet-18 and Resnet-50 using Riemannian SGD with our fixed schema.Encoder and decoder manifolds are separated with each capable of learning its own curvature for better flexibility.

\ Results For Resnet-18 tests, we see in table 2 that the new architectures perform better in all scenarios. The smallest performance was mainly seen between the hybrid models, this could generally be because the hyperbolic bias played by the additional hyperbolic components is not as prominent as in a fully hyperbolic model. This could lead to the model benefiting less from the proposed changes. We can verify this through the bigger gap in performance between the fully hyperbolic models where our proposed model sees a 74% lift in accuracy and even matches the hybrid encoders in this scenario. To study this we looked at the new curvature learned by the encoder and found that it tended towards approximately -0.6 resulting in a flatter manifold.

\ As for the Resnet-50 tests, we see in table 3, that HECNN+ is now able to greatly outperform the Euclidean model across all datasets as well. Even in the case of Tiny-Imagenet where other model accuracies begin to break down. This is probably due to the more fluid integration of hyperbolic elements and the extensive scaling to help deal with higher dimensional embeddings.

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\ Ablation We test the effect of individual model components in table 5. Each subsequent model involves the default architecture presented in the experimental setup minus the mentioned component. As we can see, the best results are achieved when all the architectural components are included. In the case of attempting to learn the curvature without the proposed optimizer schema, the model breaks completely down due to excessive numerical inaccuracies. One other benefit that we find from learning the curvature is quicker convergence. The model is able to reach convergence in 130 epochs vs the 200 epochs required by a static curve model.

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\ We then study the effectiveness of our efficient convolution in table 4. We see a ∼ 48% reduction in memory usage and ∼ 66% reduction in runtime. We attribute this improvement to the efficient closed-source convolution operations we can now leverage. However, there is still much room for improvement compared to the euclidean model. We identify the batchnorm operation as the new memory and runtime bottleneck accounting for around 60% of the runtime and around 30% of the memory. Factorizing the many parallel transports and tangent mappings required for this operation would be the next step in mitigating this issue.

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5 Conclusion

In our work, we present many new components and schemas for the use of hyperbolic deep learning in hyperbolic vision. We present a new optimizer schema that allows for curvature learning, a tanh scaling to prevent numerical precision issues, a Riemannian Adam Optimizer, and efficient formulations of existing convolutional operations. We test these components in two different problem scenarios, hierarchical metric learning and classification, and prove the potential of these new components even in float16 conditions which are notoriously unstable for hyperbolic models.

\ However, there is still much progress to be made. The scaling operations provide a general method of keeping the embeddings within the representative radius but it could also be used for norm clipping. A study has to be done on the effect of embedding bounding for hyperbolic models as it has shown to be beneficial before [14, 20]. Additionally, more efficiency can still be gained from hyperbolic models through further optimizations to the batch normalization layers. Finally, there is still the issue of the hyperbolic feedforward layer when going from higher to lower dimensionality. We currently match norms to ensure a rotation operation but we encourage finding alternate approaches that are better conforming to the manifold mathematics.

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:::info Authors:

(1) Ahmad Bdeir, Data Science Department, University of Hildesheim ([email protected]);

(2) Niels Landwehr, Data Science Department, University of Hildesheim ([email protected]).

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:::info This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.

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