Classical and quantum algorithms for scaling problems

This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

Improved Quantum Lower and Upper Bounds for Matrix Scaling

International audienc

Interior-point methods on manifolds: theory and applications

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.Comment: 85 pages. v2: Merged with independent work arXiv:2212.10981 by Hiroshi Hira

Technology-Assisted Rehabilitation of Writing Skills in Parkinson’s Disease: Visual Cueing versus Intelligent Feedback

Recent research showed that visual cueing can have both beneficial and detrimental effects on handwriting of patients with Parkinson’s disease (PD) and healthy controls depending on the circumstances. Hence, using other sensory modalities to deliver cueing or feedback may be a valuable alternative. Therefore, the current study compared the effects of short-term training with either continuous visual cues or intermittent intelligent verbal feedback. Ten PD patients and nine healthy controls were randomly assigned to one of these training modes. To assess transfer of learning, writing performance was assessed in the absence of cueing and feedback on both trained and untrained writing sequences. The feedback pen and a touch-sensitive writing tablet were used for testing. Both training types resulted in improved writing amplitudes for the trained and untrained sequences. In conclusion, these results suggest that the feedback pen is a valuable tool to implement writing training in a tailor-made fashion for people with PD. Future studies should include larger sample sizes and different subgroups of PD for long-term training with the feedback pen

Analysing empowerment-oriented email consultation for parents:development of the Guiding the Empowerment Process model

Online consultation is increasingly offered by parenting practitioners, but it is not clear if it is feasible to provide empowerment-oriented support in a single session email consultation. Based on the empowerment theory, we developed the Guiding the Empowerment Process model (GEP model) to evaluate text-based consultation. By content analysis of the email advice (n = 129; 5997 sentences in total), we investigated the feasibility of the newly developed model (inter-observer agreement, internal consistency and factor structure) and its validity. Concurrent validity was evaluated by comparing coding results using the GEP model and a Social Support model that partially intersects with empowerment. Results showed good inter-observer reliability and internal consistency of the GEP model. The results provided evidence for its concurrent validity by a significant correlation of the coding results from the GEP model with the Social Support model, although it was also distinctive. All described techniques which practitioners may employ to guide the parental process towards empowerment were observed in the sample. Also, guidance was provided in all components of the empowerment process. Feasibility of the GEP model for content analysis of email consultation in parental support from a theoretical empowerment perspective has been demonstrated

Everyday walking with Parkinson's disease: understanding personal challenges and strategies

PURPOSE: This qualitative study was designed to explore the personal experience of everyday walking with Parkinson's disease (PD), the challenges and the strategies employed to compensate for difficulties, to help contextualise the scientific knowledge base. METHODS: Semi-structured interviews were undertaken with a sample of 20 people with idiopathic PD (12 male, 8 female; mean age 65 years (range 50 - 80); mean disease duration 10 years (range 2.5 - 26). Verbatim interview transcripts were analyzed thematically using NUD*IST N6 qualitative data analysis software. RESULTS: Walking was invariably performed as an integral part of a purposeful activity within a specific context, termed walking 'plus', with challenges encountered by people with PD in three main areas: Undertaking tasks; negotiating environments; and making transitions to walking. The two key strategies to compensate for difficulties experienced were monitoring through the use of concentration, and correcting through generating rhythm and size of steps. Carers supported monitoring and correcting. CONCLUSION: People with PD need to constantly assess and drive their walking performance. Attentional resources, which can themselves be compromised in PD, were used to accomplish what is normally a largely automatic activity. Personal accounts support scientific hypotheses. Rehabilitation interventions and measurements in PD need to reflect both the physical and psychosocial context of everyday walking

Does cueing training improve physical activity in patients with Parkinson's disease?

Patients with Parkinson’s disease (PD) are encouraged to stay active to maintain their mobility. Ambulatory activity monitoring (AM) provides an objective way to determine type and amount of gait-related daily activities. Objective To investigate the effects of a home cueing training program on functional walking activity in PD. Methods In a single-blind, randomized crossover trial, PD patients allocated to early intervention received cueing training for 3 weeks, whereas the late intervention group received training in the following 3 weeks. Training was applied at home, using a prototype cueing device. AM was applied at baseline, 3, 6, and 12 weeks in the patient’s home, to record body movements. Postures and motions were classified as percentage of total time spent on (a) static activity, further specified as % sitting and % standing, and (b) % dynamic activity, further specified as % walking, % walking periods exceeding 5 seconds (W>5s) and 10 seconds (W>10s). Random coefficient analysis was applied. Results A total of 153 patients participated in this trial. Significant improvements were found for dynamic activity ( = 4.46; P 5s ( = 2.63; P 10s ( = 2.90; P < .01). All intervention effects declined significantly at 6 weeks follow-up. Conclusion Cueing training in PD patients’ own home significantly improves the amount of walking as recorded by AM. Treatment effects reduced after the intervention period, pointing to the need for permanent cueing devices and follow-up cueing training