The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces

Improving embedding of graphs with missing data by soft manifolds

Embedding graphs in continous spaces is a key factor in designing and developing algorithms for automatic information extraction to be applied in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings directly depends on how much the geometry of the continuous space matches the graph structure. Manifolds are mathematical structure that can enable to incorporate in their topological spaces the graph characteristics, and in particular nodes distances. State-of-the-art of manifold-based graph embedding algorithms take advantage of the assumption that the projection on a tangential space of each point in the manifold (corresponding to a node in the graph) would locally resemble a Euclidean space. Although this condition helps in achieving efficient analytical solutions to the embedding problem, it does not represent an adequate set-up to work with modern real life graphs, that are characterized by weighted connections across nodes often computed over sparse datasets with missing records. In this work, we introduce a new class of manifold, named soft manifold, that can solve this situation. In particular, soft manifolds are mathematical structures with spherical symmetry where the tangent spaces to each point are hypocycloids whose shape is defined according to the velocity of information propagation across the data points. Using soft manifolds for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets. Experimental results on reconstruction tasks on synthetic and real datasets show how the proposed approach enable more accurate and reliable characterization of graphs in continuous spaces with respect to the state-of-the-art

Anti-HMGCR myopathy: barriers to prompt recognition

Anti-HMGCR (3-hydroxy-3-methylglutaryl coenzyme A reductase) myopathy is an immune-mediated necrotising myopathy. Atypical presentations hinder its recognition and its prompt treatment. We present two patients with atypical clinical or pathological features. A 45-year-old woman had an asymptomatic serum creatine kinase (CK) of ~10 000 IU/L and muscle biopsy showing minimal changes. She then developed slowly progressive proximal weakness, diagnosed as limb-girdle muscular dystrophy but with negative genetics. Twelve years later, now with severe proximal weakness, her MR scan of muscle showed diffuse asymmetrical fatty degeneration, with conspicuous hyperintense STIR signal abnormalities. HMGCR antibodies were positive and she partially improved with immunosuppression. The second patient developed slowly progressive proximal limb weakness with a high serum CK (~4000 IU/L); muscle biopsy showed a lymphocyte infiltrate with angiocentric distribution suggesting vasculitis. Serum HMGCR antibodies were positive. Anti-HMGCR myopathy can present as a slowly progressive myopathy with atypical pathology. HMGCR antibody screening is indicated for people with suspected limb-girdle muscular dystrophy or atypical inflammatory muscle conditions

Genetic Modifiers of Duchenne Muscular Dystrophy and Dilated Cardiomyopathy.

OBJECTIVE: Dilated cardiomyopathy (DCM) is a major complication and leading cause of death in Duchenne muscular dystrophy (DMD). DCM onset is variable, suggesting modifier effects of genetic or environmental factors. We aimed to determine if polymorphisms previously associated with age at loss of independent ambulation (LoA) in DMD (rs28357094 in the SPP1 promoter, rs10880 and the VTTT/IAAM haplotype in LTBP4) also modify DCM onset. METHODS: A multicentric cohort of 178 DMD patients was genotyped by TaqMan assays. We performed a time-to-event analysis of DCM onset, with age as time variable, and finding of left ventricular ejection fraction \u3c 50% and/or end diastolic volume \u3e 70 mL/m2 as event (confirmed by a previous normal exam \u3c 12 months prior); DCM-free patients were censored at the age of last echocardiographic follow-up. RESULTS: Patients were followed up to an average age of 15.9 ± 6.7 years. Seventy-one/178 patients developed DCM, and median age at onset was 20.0 years. Glucocorticoid corticosteroid treatment (n = 88 untreated; n = 75 treated; n = 15 unknown) did not have a significant independent effect on DCM onset. Cardiological medications were not administered before DCM onset in this population. We observed trends towards a protective effect of the dominant G allele at SPP1 rs28357094 and recessive T allele at LTBP4 rs10880, which was statistically significant in steroid-treated patients for LTBP4 rs10880 (\u3c 50% T/T patients developing DCM during follow-up [n = 13]; median DCM onset 17.6 years for C/C-C/T, log-rank p = 0.027). CONCLUSIONS: We report a putative protective effect of DMD genetic modifiers on the development of cardiac complications, that might aid in risk stratification if confirmed in independent cohorts

Genetic modifiers of Duchenne muscular dystrophy and dilated cardiomyopathy

OBJECTIVE: Dilated cardiomyopathy (DCM) is a major complication and leading cause of death in Duchenne muscular dystrophy (DMD). DCM onset is variable, suggesting modifier effects of genetic or environmental factors. We aimed to determine if polymorphisms previously associated with age at loss of independent ambulation (LoA) in DMD (rs28357094 in the SPP1 promoter, rs10880 and the VTTT/IAAM haplotype in LTBP4) also modify DCM onset. METHODS: A multicentric cohort of 178 DMD patients was genotyped by TaqMan assays. We performed a time-to-event analysis of DCM onset, with age as time variable, and finding of left ventricular ejection fraction 70 mL/m2 as event (confirmed by a previous normal exam < 12 months prior); DCM-free patients were censored at the age of last echocardiographic follow-up. RESULTS: Patients were followed up to an average age of 15.9 \ub1 6.7 years. Seventy-one/178 patients developed DCM, and median age at onset was 20.0 years. Glucocorticoid corticosteroid treatment (n = 88 untreated; n = 75 treated; n = 15 unknown) did not have a significant independent effect on DCM onset. Cardiological medications were not administered before DCM onset in this population. We observed trends towards a protective effect of the dominant G allele at SPP1 rs28357094 and recessive T allele at LTBP4 rs10880, which was statistically significant in steroid-treated patients for LTBP4 rs10880 (< 50% T/T patients developing DCM during follow-up [n = 13]; median DCM onset 17.6 years for C/C-C/T, log-rank p = 0.027). CONCLUSIONS: We report a putative protective effect of DMD genetic modifiers on the development of cardiac complications, that might aid in risk stratification if confirmed in independent cohorts

Benefits of glucocorticoids in non-ambulant boys/men with Duchenne muscular dystrophy: A multicentric longitudinal study using the Performance of Upper Limb test

The aim of this study was to establish the possible effect of glucocorticoid treatment on upper limb function in a cohort of 91 non-ambulant DMD boys and adults of age between 11 and 26 years. All 91 were assessed using the Performance of Upper Limb test. Forty-eight were still on glucocorticoid after loss of ambulation, 25 stopped steroids at the time they lost ambulation and 18 were GC naive or had steroids while ambulant for less than a year. At baseline the total scores ranged between 0 and 74 (mean 41.20). The mean total scores were 47.92 in the glucocorticoid group, 36 in those who stopped at loss of ambulation and 30.5 in the naive group (p <0.001). The 12-month changes ranged between -20 and 4 (mean -4.4). The mean changes were -3.79 in the glucocorticoid group, -5.52 in those who stopped at loss of ambulation and -4.44 in the naive group. This was more obvious in the patients between 12 and 18 years and at shoulder and elbow levels. Our findings suggest that continuing glucocorticoids throughout teenage years and adulthood after loss of ambulation appears to have a beneficial effect on upper limb function. (C) 2015 The Authors. Published by Elsevier B.V

Upper limb function in Duchenne muscular dystrophy: 24 month longitudinal data

The aim of the study was to establish 24 month changes in upper limb function using a revised version of the performance of upper limb test (PUL 2.0) in a large cohort of ambulant and non-ambulant boys with Duchenne muscular dystrophy and to identify possible trajectories of progression. Of the 187 patients studied, 87 were ambulant (age range: 7\u201315.8 years), and 90 non-ambulant (age range: 9.08\u201324.78). The total scores changed significantly over time (p&lt;0.001). Non-ambulant patients had lower total scores at baseline (mean 19.7) when compared to the ambulant ones (mean 38.4). They also had also a bigger decrease in total scores over 24 months compared to the ambulant boys (4.36 vs 2.07 points). Multivariate model analysis showed that the Performance of Upper Limb changes reflected the entry level and ambulation status, that were independently associated to the slope of Performance of Upper Limb changes. This information will be of help both in clinical practice and at the time of designing clinical trials

Reliability of the Performance of Upper Limb assessment in Duchenne muscular dystrophy

Abstract The Performance of Upper Limb was specifically designed to assess upper limb function in Duchenne muscular dystrophy. The aim of this study was to assess (1) a cohort of typically developing children from the age of 3 years onwards in order to identify the age when the activities assessed in the individual items are consistently achieved, and (2) a cohort of 322 Duchenne children and young adults to establish the range of findings at different ages. We collected normative data for the scale validation on 277 typically developing subjects from 3 to 25 years old. A full score was consistently achieved by the age of 5 years. In the Duchenne cohort there was early involvement of the proximal muscles and a proximal to distal progressive involvement. The scale was capable of measuring small distal movements, related to activities of daily living, even in the oldest and weakest patients. Our data suggest that the assessment can be reliably used in both ambulant and non ambulant Duchenne patients in a multicentric setting and could therefore be considered as an outcome measure for future trials

Circulating Biomarkers in Neuromuscular Disorders: What Is Known, What Is New

The urgent need for new therapies for some devastating neuromuscular diseases (NMDs), such as Duchenne muscular dystrophy or amyotrophic lateral sclerosis, has led to an intense search for new potential biomarkers. Biomarkers can be classified based on their clinical value into different categories: diagnostic biomarkers confirm the presence of a specific disease, prognostic biomarkers provide information about disease course, and therapeutic biomarkers are designed to predict or measure treatment response. Circulating biomarkers, as opposed to instrumental/invasive ones (e.g., muscle MRI or nerve ultrasound, muscle or nerve biopsy), are generally easier to access and less “time-consuming”. In addition to well-known creatine kinase, other promising molecules seem to be candidate biomarkers to improve the diagnosis, prognosis and prediction of therapeutic response, such as antibodies, neurofilaments, and microRNAs. However, there are some criticalities that can complicate their application: variability during the day, stability, and reliable performance metrics (e.g., accuracy, precision and reproducibility) across laboratories. In the present review, we discuss the application of biochemical biomarkers (both validated and emerging) in the most common NMDs with a focus on their diagnostic, prognostic/predictive and therapeutic application, and finally, we address the critical issues in the introduction of new biomarkers