On Machine Learning Knowledge Representation In The Form Of Partially Unitary Operator. Knowledge Generalizing Operator

12/22/2022

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by   Vladislav Gennadievich Malyshkin, et al.

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A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial 𝐼𝑁 attributes and 𝑂𝑈𝑇 class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from 𝐼𝑁 Hilbert space into 𝑂𝑈𝑇 Hilbert space is then built from an optimization problem of transferring maximal possible probability from 𝐼𝑁 to 𝑂𝑈𝑇, this leads to the formulation of a new algebraic problem. Constructed Knowledge Generalizing Operator 𝒰 can be considered as a 𝐼𝑁 to 𝑂𝑈𝑇 quantum channel; it is a partially unitary rectangular matrix of the dimension dim(𝑂𝑈𝑇) ×dim(𝐼𝑁) transforming operators as A^𝑂𝑈𝑇=𝒰 A^𝐼𝑁𝒰^†. Whereas only operator 𝒰 projections squared are observable ⟨𝑂𝑈𝑇|𝒰|𝐼𝑁⟩^2 (probabilities), the fundamental equation is formulated for the operator 𝒰 itself. This is the reason of high generalizing power of the approach; the situation is the same as for the Schrödinger equation: we can only measure ψ^2, but the equation is written for ψ itself.