Title: Implicit regularization via Spectral Neural Networks and non-linear matrix sensing. Abstract: The phenomenon of \textit{implicit regularization} has attracted interest in the recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient flow dynamics in many neural nets, even without any explicit regularizer in the loss function, converges to the solution of a regularized learning problem. However, known results attempting to theoretically explain this phenomenon focus overwhelmingly on the setting of linear neural nets, and the simplicity of the linear structure is particularly crucial to existing arguments. In this paper, we explore this problem in the context of more realistic neural networks with a general class of non-linear activation functions, and rigorously demonstrate the implicit regularization phenomenon for such networks in the setting of matrix sensing problems. This is coupled with rigorous rate guarantees that ensure exponentially fast convergence of gradient descent, complemented by matching lower bounds which stipulate that the exponential rate is the best achievable. In this vein, we contribute a network architecture called Spectral Neural Networks (\textit{abbrv.} SNN) that is particularly suitable for matrix learning problems. Conceptually, this entails coordinatizing the space of matrices by their singular values and singular vectors, as opposed to by their entries, a potentially fruitful perspective for matrix learning. We demonstrate that the SNN architecture is inherently much more amenable to theoretical analysis than vanilla neural nets and confirm its effectiveness in the context of matrix sensing, supported via both mathematical guarantees and empirical investigations. We believe that the SNN architecture has the potential to be of wide applicability in a broad class of matrix learning scenarios.