Geometric representations for minimalist grammars

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations.Comment: 43 pages, 4 figure

Efficient approximation of probability distributions with k-order decomposable models

During the last decades several learning algorithms have been proposed to learn probability distributions based on decomposable models. Some of these algorithms can be used to search for a maximum likelihood decomposable model with a given maximum clique size, k. Unfortunately, the problem of learning a maximum likelihood decomposable model given a maximum clique size is NP-hard for $k<2$. In this work, we propose the fractal tree family of algorithms which approximates this problem with a computational complexity of $\mathcal{O}(k \cdot n^2 \log n)$ in the worst case, where $n$ is the number of implied random variables and N is the size of the training set. The fractal tree algorithms construct a sequence of maximal $i$-order decomposable graphs, for $i=2,...,k,$ in $k - 1$ steps. At each step, the algorithms follow a divide-and-conquer strategy that decomposes the problem into a set of separate problems. Each separate problem is efficiently solved using the generalized Chow-Liu algorithm. Fractal trees can be considered a natural extension of the Chow-Liu algorithm, from $k = 2$ to arbitrary values of $k$, and they have shown a competitive behavior to deal with the maximum likelihood problem. Due to their competitive behavior, their low computational complexity and their modularity, which allow them to implement different parallelization strategies, the proposed procedures are especially advisable for modeling high dimensional domains

A New General Allometric Biomass Model

To implement monitoring and assessment of national forest biomass, it is becoming the trend to develop generalized single-tree biomass models suitable for large scale forest biomass estimation. Considering that the theoretical biomass allometric model developed by West et al. [1,2] was statistically different from the empirical one, the two parameters in the most commonly used biomass equation M=aDb were analyzed in this paper. Firstly, based on the knowledge of geometry, the theoretical value of parameter b was deduced, i.e., b=7/3(~2.33), and the comparison with many empirical studies conducted throughout the globe indicated that the theoretical parameter could describe soundly the average allometric relationship between aboveground biomass M and D (diameter on breast height). Secondly, using five datasets of aboveground biomass which consisted of 1441 M-D pairs of sample trees, the new general biomass allometric model was validated. Finally, the relationship between parameter a and wood density p was analyzed, and the linear regression was developed. The new model, which is not only simple but also species-specific, offers a feasible approach on establishment of generalized biomass models for regional and national forest biomass estimation