Spectral methods: crucial for machine learning, natural for quantum computers?
This is a speculative proposal for leveraging quantum computing to potentially revolutionize spectral methods in machine learning, but it is incremental as it builds on existing quantum and classical ideas without empirical validation.
The article argues that quantum computers could enable new machine learning methods by naturally implementing spectral techniques, such as manipulating Fourier spectra via the Quantum Fourier Transform, which are fundamental to many classical ML approaches like deep learning and CNNs.
This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.