Trading Positional Complexity vs. Deepness in Coordinate Networks
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866918384820551680 |
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| author | Zheng, Jianqiao Ramasinghe, Sameera Li, Xueqian Lucey, Simon |
| author_facet | Zheng, Jianqiao Ramasinghe, Sameera Li, Xueqian Lucey, Simon |
| contents | It is well noted that coordinate-based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been mainly studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. In addition, we argue that employing a more complex positional encoding -- that scales exponentially with the number of modes -- requires only a linear (rather than deep) coordinate function to achieve comparable performance. Counter-intuitively, we demonstrate that trading positional embedding complexity for network deepness is orders of magnitude faster than current state-of-the-art; despite the additional embedding complexity. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_08987 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Trading Positional Complexity vs. Deepness in Coordinate Networks Zheng, Jianqiao Ramasinghe, Sameera Li, Xueqian Lucey, Simon Computer Vision and Pattern Recognition It is well noted that coordinate-based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been mainly studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. In addition, we argue that employing a more complex positional encoding -- that scales exponentially with the number of modes -- requires only a linear (rather than deep) coordinate function to achieve comparable performance. Counter-intuitively, we demonstrate that trading positional embedding complexity for network deepness is orders of magnitude faster than current state-of-the-art; despite the additional embedding complexity. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. |
| title | Trading Positional Complexity vs. Deepness in Coordinate Networks |
| topic | Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2205.08987 |