Intertwining wavelets or Multiresolution analysis on graphs through random forests

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure

Proteomic analysis of heart failure hospitalization among patients with chronic kidney disease: The Heart and Soul Study.

BACKGROUND:Patients with chronic kidney disease (CKD) are at increased risk for heart failure (HF). We aimed to investigate differences in proteins associated with HF hospitalizations among patients with and without CKD in the Heart and Soul Study. METHODS AND RESULTS:We measured 1068 unique plasma proteins from baseline samples of 974 participants in The Heart and Soul Study who were followed for HF hospitalization over a median of 7 years. We sequentially applied forest regression and Cox survival analyses to select prognostic proteins. Among participants with CKD, four proteins were associated with HF at Bonferroni-level significance (p<2.5x10(-4)): Angiopoietin-2 (HR[95%CI] 1.45[1.33, 1.59]), Spondin-1 (HR[95%CI] 1.13 [1.06, 1.20]), tartrate-resistant acid phosphatase type 5 (HR[95%CI] 0.65[0.53, 0.78]) and neurogenis locus notch homolog protein 1 (NOTCH1) (HR[95%CI] 0.67[0.55, 0.80]). These associations persisted at p<0.01 after adjustment for age, estimated glomerular filtration and history of HF. CKD was a significant interaction term in the associations of NOTCH1 and Spondin-1 with HF. Pathway analysis showed a trend for higher representation of the Cardiac Hypertrophy and Complement/Coagulation pathways among proteins prognostic of HF in the CKD sub-group. CONCLUSIONS:These results suggest that markers of heart failure differ between patients with and without CKD. Further research is needed to validate novel markers in cohorts of patients with CKD and adjudicated HF events

Random Forests and Networks Analysis

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.Comment: Survey pape

Prediction of non-genotoxic carcinogenicity based on genetic profiles of short term exposure assays

Non-genotoxic carcinogens are substances that induce tumorigenesis by non-mutagenic mechanisms and long term rodent bioassays are required to identify them. Recent studies have shown that transcription profiling can be applied to develop early identifiers for long term phenotypes. In this study, we used rat liver expression profiles from the NTP (National Toxicology Program, Research Triangle Park, USA) DrugMatrix Database to construct a gene classifier that can distinguish between non-genotoxic carcinogens and other chemicals. The model was based on short term exposure assays (3 days) and the training was limited to oxidative stressors, peroxisome proliferators and hormone modulators. Validation of the predictor was performed on independent toxicogenomic data (TG-GATEs, Toxicogenomics Project-Genomics Assisted Toxicity Evaluation System, Osaka, Japan). To build our model we performed Random Forests together with a recursive elimination algorithm (VarSelRF). Gene set enrichment analysis was employed for functional interpretation. A total of 770 microarrays comprising 96 different compounds were analyzed and a predictor of 54 genes was built. Prediction accuracy was 0.85 in the training set, 0.87 in the test set and increased with increasing concentration in the validation set: 0.6 at low dose, 0.7 at medium doses and 0.81 at high doses. Pathway analysis revealed gene prominence of cellular respiration, energy production and lipoprotein metabolism. The biggest target of toxicogenomics is accurately predict the toxicity of unknown drugs. In this analysis, we presented a classifier that can predict non-genotoxic carcinogenicity by using short term exposure assays. In this approach, dose level is critical when evaluating chemicals at early time points.Fil: Perez, Luis Orlando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Centro Nacional Patagónico. Instituto Patagónico para el Estudio de los Ecosistemas Continentales; ArgentinaFil: González José, Rolando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Centro Nacional Patagónico. Instituto Patagónico para el Estudio de los Ecosistemas Continentales; ArgentinaFil: Peral Garcia, Pilar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico CONICET- La Plata. Instituto de Genética Veterinaria ; Argentin

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure