Competitive Object Learning

In addition to subsampling, goals of competitive object learning are the minimization of the expected quantization error and entropy maximization. A finite set of 3D scan points $\cal D$ is subsambled to the set ${\cal A} = \{\V w_1, \V w_2, \ldots, \V w_N\}$. Error minimization is done with respect to the following function:

$$E ( {\cal D}, {\cal A} ) = \frac{1}{\vert{\cal D}\vert} \sum_{\V w_i \in {\cal A}} \sum_{\V \xi \in R_c} \norm {\V \xi - \V w_i},$$

with the set ${\cal A}$ of samples and the Voronoi set ${\cal R}_c$ of unit $c$, i.e., ${\cal R}_c = \{ \V \xi \in {\cal D} \vert s(\V \xi) = c \}$ and $s(\V \xi) = \arg \min_{c \in {\cal A}} \norm {\V \xi - \V w_c}$. Entropy maximization guarantees inherent robustness. The failure of reference vectors, i.e., missing 3D points, affects only a limited fraction of the data. Interpreting the generation of an input signal and the subsequent mapping onto the nearest sample in ${\cal A}$ as a random experiment which assigns a value $x \in {\cal A}$ to the random variable $X$, then maximizing the entropy

$$H(X) = - \sum_{x \in {\cal A}} P(x) \log(P(x))$$

is equivalent to equiprobable samples. The following neural gas algorithm learns and subsamples 3D points clouds [7]:

i.).

Initialize the set $\cal A$ to contain $N$ vectors, randomly from the input set. Set $t = 0$.

ii.).

Generate at random an input element $\V \xi$, i.e., select a point from ${\cal D}$.

iii.).

Order all elements of $\cal A$ according to their distance to $\V \xi$, i.e., find the sequence of indices $(i_0, i_1, \ldots, i_{N-1})$ such that $\V w_{i_0}$ is the reference vector closest to $\V \xi$, $\V w_{i_1}$ is the reference vector second closest to $\V \xi$, etc., $\V w_{i_k}$, $k = 0, \ldots, N-1$ is the reference vector such that $k$ vectors $\V w_j$ exists that are closer to $\V \xi$ than $\V w_{i_k}$. $k_i(\V \xi, {\cal A})$ denotes the number $k$ associated with $\V w_i$.

iv.).

Adapt the reference vectors according to

$$\Delta \V w_i = \varepsilon(t) h_\lambda(k_i(\V \xi, {\cal A})) \cdot (\V \xi - \V w_i),$$

with the following time dependencies:

$$\lambda ( t) = \lambda_i (\lambda_f / \lambda_i)^{t/t_\text{max}},$$

$$end{tex2html_deferred}$$

$$end{tex2html_deferred} \varepsilon ( t) = \varepsilon_i (\varepsilon_f / \varepsilon_i)^{t/t_\text{max}},$$

$$end{tex2html_deferred}$$

$$end{tex2html_deferred} h_\lambda (k) = \exp(-k/\lambda(t)).$$

v.).

Increase the time parameter $t$.

The neural gas algorithms is used with the following parameters: $\lambda_f = 0.01$, $\lambda_i = 10.0$, $\varepsilon_i = 0.5$, $\varepsilon_f = 0.005$, $t_$max$= 10000$. Note that $t_$max controls the run time. Fig. shows 3D models of the database (top row) and subsampled versions (bottom) with 250 points.

Figure: Top: 3D models (point clouds) of the database. Bottom: sumbsampled models with 250 points.