Bi-stochastic kernels via asymmetric affinity functions

In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function {\alpha}. The affinity function {\alpha} measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3: Minor changes and edits. v4: Edited comments and added DO

Non-linear nitrogen dynamics and calcium depletion along a temperate forest soil nitrogen gradient

Understanding how N availability influences base cation stores is critical for long-term ecosystem sustainability. Indices of nitrogen (N) availability and the distribution of nutrients in plant biomass, soil, and soil water were examined across ten young, unpolluted Douglas-fir (Pseudotsuga menziesii) stands in the Oregon Coast Range spanning a three-fold soil N gradient (0-10 cm: 0.21 - 0.69% N, 0-100 cm: 9.2 – 28.8 Mg N . ha⁻¹) but having similar stand age and sandstone parent material. δ¹⁵N in foliage and forest floor increased across the gradient and approached the isotopic signature of the atmosphere at high soil N stands, suggesting that variation in N accumulation across sites is related to historic site occupancy by N₂-fixing red alder (Alnus rubra). Although no longer present on these sites, red alder stands can add 100-200 kg N ha⁻¹ yr⁻¹ to an ecosystem for decades, a significantly higher N input than precipitation (0.65 kg N ha⁻¹ yr⁻¹). Annual net N mineralization and litterfall N return displayed non-linear relationships with soil N, increasing initially, and then decreasing at more N-rich sites. In contrast, nitrate leaching from deep soils increased linearly across the soil N gradient and ranged from 0.074 to 30 kg N . ha⁻¹ . yr⁻¹. Nitrogen availability was negatively correlated with indices of Ca availability. Soil exchangeable Ca, Mg, and K pools to 1 m depth were negatively related to nitrate losses across sites. Calcium was the only base cation that decreased in both plant and soil pools across the soil N gradient, and a greater proportion of total available ecosystem Ca was sequestered in plant biomass at high N, low Ca sites. The preferential storage of Ca in aboveground biomass at high N and low Ca sites, while critical for sustaining plant productivity, may also predispose forests to Ca depletion in areas managed for intensive biomass removal. Our work supports a hierarchical model of coupled N-Ca cycles across gradients of soil N enrichment, with microbial production of mobile nitrate leading to depletion of readily available Ca at the ecosystem scale, and plant sequestration promoting Ca conservation as Ca supply diminishes. Long-term N enrichment of temperate forest soils appears capable of sustaining an open N cycle and key symptoms of N saturation for multiple decades after the cessation of elevated N inputs

A Sharpened Condition for Strict Log-Convexity of the Spectral Radius via the Bipartite Graph

Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(e^D A) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(e^D A) is strictly convex over D_1, D_2 if and only if D_1-D_2 != c I for any c \in R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and A'A be irreducible, which is the sharpest possible replacement condition. Irreducibility of both A and A'A is shown to be equivalent to irreducibility of A^2 and A'A, which is the condition for a number of strict inequalities on the spectral radius found in Cohen, Friedland, Kato, and Kelly (1982). Such `two-fold irreducibility' is equivalent to joint irreducibility of A, A^2, A'A, and AA', or in combinatorial terms, equivalent to the directed graph of A being strongly connected and the simple bipartite graph of A being connected. Additional ancillary results are presented.Comment: 20 pages; v. 2: expanded exposition

Gradient methods for problems with inexact model of the objective

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments

Auto-labelling of Markers in Optical Motion Capture by Permutation Learning

Optical marker-based motion capture is a vital tool in applications such as motion and behavioural analysis, animation, and biomechanics. Labelling, that is, assigning optical markers to the pre-defined positions on the body is a time consuming and labour intensive postprocessing part of current motion capture pipelines. The problem can be considered as a ranking process in which markers shuffled by an unknown permutation matrix are sorted to recover the correct order. In this paper, we present a framework for automatic marker labelling which first estimates a permutation matrix for each individual frame using a differentiable permutation learning model and then utilizes temporal consistency to identify and correct remaining labelling errors. Experiments conducted on the test data show the effectiveness of our framework

Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers

Building on recent progress at the intersection of combinatorial optimization and deep learning, we propose an end-to-end trainable architecture for deep graph matching that contains unmodified combinatorial solvers. Using the presence of heavily optimized combinatorial solvers together with some improvements in architecture design, we advance state-of-the-art on deep graph matching benchmarks for keypoint correspondence. In addition, we highlight the conceptual advantages of incorporating solvers into deep learning architectures, such as the possibility of post-processing with a strong multi-graph matching solver or the indifference to changes in the training setting. Finally, we propose two new challenging experimental setups. The code is available at https://github.com/martius-lab/blackbox-deep-graph-matchingComment: ECCV 2020 conference pape

Forest calcium depletion and biotic retention along a soil nitrogen gradient

High nitrogen (N) accumulation in terrestrial ecosystems can shift patterns of nutrient limitation and deficiency beyond N toward other nutrients, most notably phosphorus (P) and base cations (calcium [Ca], magnesium [Mg], and potassium [K]). We examined how naturally high N accumulation from a legacy of symbiotic N fixation shaped P and base cation cycling across a gradient of nine temperate conifer forests in the Oregon Coast Range. We were particularly interested in whether long-term legacies of symbiotic N fixation promoted coupled N and organic P accumulation in soils, and whether biotic demands by non-fixing vegetation could conserve ecosystem base cations as N accumulated. Total soil N (0–100 cm) pools increased nearly threefold across the N gradient, leading to increased nitrate leaching, declines in soil pH from 5.8 to 4.2, 10-fold declines in soil exchangeable Ca, Mg, and K, and increased mobilization of aluminum. These results suggest that long-term N enrichment had acidified soils and depleted much of the readily weatherable base cation pool. Soil organic P increased with both soil N and C across the gradient, but soil inorganic P, biomass P, and P leaching loss did not vary with N, implying that historic symbiotic N fixation promoted soil organic P accumulation and P sufficiency for non-fixers. Even though soil pools of Ca, Mg, and K all declined as soil N increased, only Ca declined in biomass pools, suggesting the emergence of Ca deficiency at high N. Biotic conservation and tight recycling of Ca increased in response to whole-ecosystem Ca depletion, as indicated by preferential accumulation of Ca in biomass and surface soil. Our findings support a hierarchical model of coupled N–Ca cycling under long-term soil N enrichment, whereby ecosystem-level N saturation and nitrate leaching deplete readily available soil Ca, stimulating biotic Ca conservation as overall supply diminishes. We conclude that a legacy of biological N fixation can increase N and P accumulation in soil organic matter to the point that neither nutrient is limiting to subsequent non-fixers, while also resulting in natural N saturation that intensifies base cation depletion and deficiency.Keywords: Potassium, Base cation depletion, Temperate forest, Calcium, Magnesium, Douglas-fir, Aluminum, Phosphorus, Nitrate leaching, Nitrogen saturatio