Learning gradients on manifolds

A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on manifolds for dimension reduction for high-dimensional data with few observations. We obtain generalization error bounds for the gradient estimates and show that the convergence rate depends on the intrinsic dimension of the manifold and not on the dimension of the ambient space. We illustrate the efficacy of this approach empirically on simulated and real data and compare the method to other dimension reduction procedures.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ206 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Heritability of Plant Architecture in Diploid Roses (Rosa spp.)

Plant architecture is very important because it helps understand the plant organization and the interaction between the plant and the environment. In the preliminary study with four F1 families, 13 architectural traits were evaluated, and six traits were calculated in May of 2014 in College Station, TX to estimate variability, phenotypic correlations and principle components. All architectural traits except the length of secondary vegetative part, the length of secondary shoots, the branching angle between primary and secondary shoots and the internode length of both order level shoots differed among the four rose populations. The same traits on different order level shoots were generally correlated as were some of the different traits. The most common inflorescence structure type observed among rose seedlings was a cyme, although other types such as a reversed raceme, raceme, solitary flower and even mixed types and unknown types were observed. Based on the result of PCA, the attribute that best explain the variability observed in our rose seedlings are the number of nodes on the secondary shoot, the length of the reproductive part and the internode length on primary and secondary shoots. By combining the preliminary data with that from previous studies, we chose six architectural traits for subsequent study. They are plant height, the number of primary shoots, the length of the primary shoots, the number of nodes on the primary shoots, the number of secondary shoots per primary shoot, and the number of tertiary shoots per primary shoot. In 2015, six rose plant architectural traits were evaluated in May and December in College Station, TX to estimate variability and heritability. Most traits showed a substantial amount of variability. A random effects model Restricted Maximum Likelihood (REML) analysis was used to estimate the genetic components, narrow sense heritability and broad sense heritability. Architectural traits demonstrated low to moderate narrow sense heritability (0.12-0.50) and low to high broad sense heritability (0.25-0.92). Traits with low narrow sense heritability but moderately high to high broad sense heritability (number of primary shoots, the length of primary shoots and the number of nodes on the primary shoot) indicate an important non-additive genetic component. The number of nodes on the primary shoots, and the number of secondary and tertiary shoots per primary shoot were greatly affected by the genotype by environment interaction. Most families, except for the three with ‘Vineyard Song’ as a parent, did not increase in the number of nodes on the primary shoot over the season. In contrast, 11 out of 13 families had more secondary and tertiary shoots form during the year. Even among those with increased numbers of secondary and tertiary shoots, the number varied among families. For these traits selection would need to be done in both seasons whereas with plant height, shoot length and the number of primary shoots selection in either the early or late season would be effective. A comparison of desirable and undesirable plant growth types indicated that the key differences were in the number of primary shoots and in the density of secondary/tertiary shoots on the primary shoot, with more desirable types having more than thirty primary shoots with multiple secondary/tertiary shoots