Jonathan H. Manton

Optimization algorithms exploiting unitary constraints

This paper presents novel algorithms that iteratively converge to a local minimum of a real-valued function f (X) subject to the constraint that the columns of the complex-valued matrix X are mutually orthogonal and have unit norm. The algorithms are derived by reformulating the constrained optimization problem as an unconstrained one on a suitable manifold. This significantly reduces the dimensionality of the optimization problem. Pertinent features of the proposed framework are illustrated by using the framework to derive an algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a Hermitian matrix.

Blind source-separation using second-order cyclostationary statistics

This paper studies the blind source separation (BSS) problem with the assumption that the source signals are cyclostationary. Identifiability and separability criteria based on second-order cyclostationary statistics (SOCS) alone are derived. The identifiability condition is used to define an appropriate contrast function. An iterative algorithm (ATH2) is derived to minimize this contrast function. This algorithm separates the sources even when they do not have distinct cycle frequencies.

Persistence of skin-resident memory T cells within an epidermal niche.

Significance Tissue-resident memory T cells (TRM) form in the skin where they are retained and can protect against subsequent infection. Using a combination of intravital imaging and mathematical modeling of skin TRM that form after cutaneous herpes simplex virus 1 infection, we reveal that these memory T cells persist at the site of infection for the life of a mouse owing to slow random migration. We also report that TRM compete with dendritic epidermal γδ T cells in skin for local survival signals, suggesting that T cells compete for space within an epidermal niche. Barrier tissues such as the skin contain various populations of immune cells that contribute to protection from infections. These include recently identified tissue-resident memory T cells (TRM). In the skin, these memory CD8+ T cells reside in the epidermis after being recruited to this site by infection or inflammation. In this study, we demonstrate prolonged persistence of epidermal TRM preferentially at the site of prior infection despite sustained migration. Computational simulation of TRM migration within the skin over long periods revealed that the slow rate of random migration effectively constrains these memory cells within the region of skin in which they form. Notably, formation of TRM involved a concomitant local reduction in dendritic epidermal γδ T-cell numbers in the epidermis, indicating that these populations persist in mutual exclusion and may compete for local survival signals. Accordingly, we show that expression of the aryl hydrocarbon receptor, a transcription factor important for dendritic epidermal γδ T-cell maintenance in skin, also contributes to the persistence of skin TRM. Together, these data suggest that skin tissue-resident memory T cells persist within a tightly regulated epidermal T-cell niche.

Stochastic consensus over noisy networks with Markovian and arbitrary switches

This paper considers stochastic consensus problems over lossy wireless networks. We first propose a measurement model with a random link gain, additive noise, and Markovian lossy signal reception, which captures uncertain operational conditions of practical networks. For consensus seeking, we apply stochastic approximation and derive a Markovian mode dependent recursive algorithm. Mean square and almost sure (i.e., probability one) convergence analysis is developed via a state space decomposition approach when the coefficient matrix in the algorithm satisfies a zero row and column sum condition. Subsequently, we consider a model with arbitrary random switching and a common stochastic Lyapunov function technique is used to prove convergence. Finally, our method is applied to models with heterogeneous quantizers and packet losses, and convergence results are proved.

The geometry of weighted low-rank approximations

The low-rank approximation problem is to approximate optimally, with respect to some norm, a matrix by one of the same dimension but smaller rank. It is known that under the Frobenius norm, the best low-rank approximation can be found by using the singular value decomposition (SVD). Although this is no longer true under weighted norms in general, it is demonstrated here that the weighted low-rank approximation problem can be solved by finding the subspace that minimizes a particular cost function. A number of advantages of this parameterization over the traditional parameterization are elucidated. Finding the minimizing subspace is equivalent to minimizing a cost function on the Grassmann manifold. A general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework. Within this framework, two novel algorithms (a steepest descent algorithm and a Newton-like algorithm) are derived for solving the weighted low-rank approximation problem. They are compared with other algorithms for low-rank approximation as well as with other algorithms for minimizing a cost function on a Grassmann manifold.

Optimal training sequences and pilot tones for OFDM systems

Orthogonal frequency-division multiplex (OFDM) systems transmit data in blocks. The two simplest ways of identifying the channel in OFDM systems are to insert a training sequence between consecutive blocks or to insert pilot tones inside each block. This article proves that both methods can achieve the same level of performance under certain conditions on the block length.

High spatial and temporal resolution wide-field imaging of neuron activity using quantum NV-diamond

A quantitative understanding of the dynamics of biological neural networks is fundamental to gaining insight into information processing in the brain. While techniques exist to measure spatial or temporal properties of these networks, it remains a significant challenge to resolve the neural dynamics with subcellular spatial resolution. In this work we consider a fundamentally new form of wide-field imaging for neuronal networks based on the nanoscale magnetic field sensing properties of optically active spins in a diamond substrate. We analyse the sensitivity of the system to the magnetic field generated by an axon transmembrane potential and confirm these predictions experimentally using electronically-generated neuron signals. By numerical simulation of the time dependent transmembrane potential of a morphologically reconstructed hippocampal CA1 pyramidal neuron, we show that the imaging system is capable of imaging planar neuron activity non-invasively at millisecond temporal resolution and micron spatial resolution over wide-fields.

Affine precoders for reliable communications

It is known that precoding a signal prior to its transmission through an unknown finite impulse response channel facilitates the equalisation of the channel. Moreover, since linear precoders spread the spectrum, the equalisation process mitigates the effects of channel spectral nulls caused by frequency selective fading. This paper argues that certain affine precoders are more efficient than linear precoders. Here, the efficiency of a precoder is a measure of its ability to enable the receiver to recover the source signal relative to the amount of redundancy introduced by the precoder. An efficient affine precoder is heuristically derived and simulations used to demonstrate that it is superior to both the traditional training sequence method and the filter bank precoding scheme. This precoding and equalisation scheme naturally extends to time varying channels.

A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.

The Geometry of the Newton Method on Non-Compact Lie Groups

An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is non-compact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.