Random Hessians and Jacobians: theory and applications

Properties of complicated 'landscapes', i.e. random
functions defined on very high dimensional spaces, have recently attracted considerable attention, e.g. in theory of Deep Machine Learning and Optimization. In particular, one may be interested in number of 'valleys' (i.e. local minima) at a given 'height', 'ridges' or barriers separating them, and more generally 'critical points' (saddles and maxima). An important role in characterising geometry of the landscapes, especially close to the critical points, is played by the matrix of second derivatives known as the Hessian. It determines e.g. the gradient descent dynamics within these landscapes, which has many practical applications for search algorithms. Depending on the context, the landscape can correspond to the energy of a physical system, to the loss function of a machine-learning algorithm, to the cost function of an optimization problem, or to
the fitness function of a biological system. In the analysis of critical points the (modulus of) the characteristic polynomial of the Hessian appears naturally. Similarly, to characterize equilibria in complicated dynamical systems (e.g. communities of many interacting species) requires investigating properties of more general, asymmetric, Jacobian matrices, for which Hessians are only a special case. Jacobians are deeply related to questions of stability of systems under small perturbations, and as such are very fundamental. Note that in contrast to Hessians whose spectra are real and eigenvectors form an orthogonal set, the Jacobians have in general complex eigenvalues and bi-orthogonal set of left and right eigenvectors. The studies of the associated 'eigenvector non-orthogonality' in random setting turn out to be relevant both for complex systems stability as well as to chaotic wave scattering and random lasing. The present research proposal is mainly centred around analysis of various properties of random matrices and operators,
mostly arising via Hessians of random landscapes, or random Jacobians of various origin.

1) The proposed research aims to benefit the whole UK population by enhancing UK competitiveness in the area of Random Matrix theory and its applications. In this way we will add to the available pool of the UK expertise and help increasing diversity of the existing UK research landscape.
In particular, we will identify and then contact researchers beyond the remit of Mathematical Sciences (e.g. Data Scientists and telecommunication engineers) to clarify if outcomes may have implications/applications for signal transmission security.

2) The project involves two Postdoctoral Research Assistants, who will both acquire sophisticated research skills at the interface between Mathematical/Theoretical Physics and Probability. These skills are highly transferable and can be applied to multitude of problems. This should both enhance the career prospects of the team members and benefit their future employers and collaborators in the academic or commercial sector.

3) 3-day workshop will be organized at KCL in Spring 2022, tentatively entitled ''From Random Matrices to Random Landscapes". Being centered around the main research topics of the present proposal, the workshop will help reviewing the state of art in the field, facilitate exchange of ideas and results and help to forge new collaborations. The impact will be amplified by attendance of PhD students and early career researchers who might be inspired to work in this area. Apart from the paprticipants, the main benefactor will be the whole KCL research community enjoying increased visibility in this area.

4) Finally, some elements of Random Matrix Theory and the landscape paradigm are suitable topics for public outreach activities, like popular science lectures, promoting the idea of an uninhibited research culture as one of the fundamental values intrinsic for contemporary civilisation and beneficial for all strata of modern society. In particular, the paradigm of resonance phenomena - the topic closely related to the proposed research - is in the core of our technology, with applications ranging from musical instruments to lasers. This may be a bridge towards introducing and visualizing important mathematical ideas like, for example, that of a ''spectrum'', which are central for the research in Random Matrices. Such a topic will ideally suit for presentations during Natural and Mathematical Sciences open days at KCL, with beneficiaries including around 200-300 16-18 year old London based school students taking part in those days. They will benefit by increasing their knowledge and awareness of the important mathematical concepts.