On Machine Learning Knowledge Representation In The Form Of Partially Unitary Operator. Knowledge Generalizing Operator
This work proposes a novel foundational approach to ML knowledge representation with potential broad impact, though it appears incremental in applying quantum-inspired methods to ML.
The paper tackles the problem of machine learning knowledge representation by developing a new form using a partially unitary operator to transform input attributes and output class labels into Hilbert spaces, achieving high generalization power through an optimization problem that maximizes probability transfer. The result is a Knowledge Generalizing Operator that acts as a quantum channel, formulated from a new algebraic problem.
A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial $\mathit{IN}$ attributes and $\mathit{OUT}$ class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from $\mathit{IN}$ Hilbert space into $\mathit{OUT}$ Hilbert space is then built from an optimization problem of transferring maximal possible probability from $\mathit{IN}$ to $\mathit{OUT}$, this leads to the formulation of a new algebraic problem. Constructed Knowledge Generalizing Operator $\mathcal{U}$ can be considered as a $\mathit{IN}$ to $\mathit{OUT}$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\mathrm{dim}(\mathit{OUT}) \times \mathrm{dim}(\mathit{IN})$ transforming operators as $A^{\mathit{OUT}}=\mathcal{U} A^{\mathit{IN}} \mathcal{U}^{\dagger}$. Whereas only operator $\mathcal{U}$ projections squared are observable $\left\langle\mathit{OUT}|\mathcal{U}|\mathit{IN}\right\rangle^2$ (probabilities), the fundamental equation is formulated for the operator $\mathcal{U}$ 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.