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- Quantum Energy Regression using Scattering Transforms. With Nicolas Poilvert and Stéphane Mallat. 2015. pdf, arXiv.
Deep learning architectures have recently reemerged with the advent of new computational resources and algorithmic improvements. They are yielding remarkable state of the art results in numerous learning tasks, primarily in computer vision for the analysis of images, speech recognition, and natural language processing [1-5]. Recently, such algorithms have begun branching into other areas, such as music  and even physics . However, the complexity of such algorithms means that they have remained essentially a black box, yielding proportionally little insight given their performance achievements, which limits their utility in fields outside of those traditionally tackled by the machine learning community and obstructs new scientific directions. My research aims to open this blackbox by utilizing tools from harmonic analysis to construct multiscale deep learning architectures amenable to mathematical analysis.
From structured data, such as time series, textures, images, EEG, ECG, MRI, and 3D scans, one can extract relevant features via filter-based analysis. Convolutional networks apply a sequence of filters and nonlinear averaging operations, which are learned for a specific task from training data. A particular task may necessitate that the networks encode certain fundamental invariant and stability properties, but they are not guaranteed by the algorithm and any such properties are an implicit byproduct of the algorithm rather than an explicit design choice.
A scattering transform  constructs similar architectures based on convolutional filters. Rather than beginning directly with task, however, the network is designed to encode geometric invariants and stability properties that are known to be present in the data. For example, translation and rotation invariance (either globally or locally) are often desired in image processing tasks [9-12], although there is no theoretical barrier to incorporating other invariants into the transform. Additionally, in many machine learning and data analysis tasks, small deformations of the data do not drastically affect the outcome; scattering networks are guaranteed to be stable such deformations, unlike their convolutional net counterparts.
More specifically, let be a structured data point, in which represents for example one-dimensional time, two-dimensional space, or three-dimensional volume, (hereafter referred to as “space”). A wavelet is a complex waveform that is well localized in both space and frequency. The frequency support of is essentially contained in a frequency ball centered at a central frequency .
The wavelet is dilated at scales and rotated by ,
Thus , and so is essentially supported in a frequency ball centered at , and dilated by a factor relative to the support of . See the figure below for an illustration of how the frequency support varies with and in two dimensions.
Wavelet coefficients of are computed via convolution for different scales and rotations. Wavelet coefficients are computed up to a maximum scale . Frequencies below are captured by a low pass filter , where is a positive rotationally symmetric function, such as a Gaussian. Its Fourier transform is essentially supported in the ball . The resulting wavelet transform of is defined as:
It contains similar frequency information as the Fourier transform of , but the multiscale/multiresolution wavelet approach yields numerous additional desirable properties not available to standard Fourier analysis.
The wavelet transform is linear. A nonlinear transform is obtained by taking the complex modulus of the wavelet coefficients:
The complex modulus computes the complex envelope of the wavelet coefficients . The nonlinear modulus therefore smooths the wavelet coefficients, pushing their frequency content into the lower frequencies.
Invariants are obtained by averaging over the relevant structures in the wavelet transform. The low pass transform,
is an averaging operator which is translation invariant up to the scale . Global translation invariance is achieved by letting , which in effect computes the global average of . The wavelet modulus coefficients are covariant to translations and rotations, but not invariant. Translation invariant features are obtained by applying the low pass filter on top of the wavelet modulus transform:
constitutes a set translation invariant features of the signal . Translation and rotation invariant features are obtained by replacing with a low pass filter over both translations and rotations . In fact, invariance to any finite group or Lie group action can be obtained through appropriately defined wavelet transforms, see  for more details. In what follows, we focus on translation invariance in order to simplify the presentation.
The low pass transform … [to be continued]
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