This page describes how weights can be specified depending on their type.

Note that the weights in the following examples are interpreted as weights elevated to a power \(p\) because, by default, the numerical values read by LatNet Builder are interpreted as the constants \(\gamma_{\mathfrak u}^p\), i.e. the weights elevated to the power \(p\) matching the choice of the \(\ell_q\)-norm. The choice of this norm is made using –norm-type q. An exception to this occurs when using the infinity-norm, where they are interpreted as if \(p = 1\). To interpret the numerical values as another power of the weights, say 1, the option

  --weights-power 1

must be added to the command line.

Projection-Dependent Weights

In that case, the value of the –weights option consists of colon-separated tokens: the first is the type of weights (projection-dependent); the other tokens are optional and assign explicit weights to projections. Projections for which no weights have been explicitly assigned have a weight equal to zero.

Projection-dependent weights with \(\gamma_{\{1,2,3\}}^p = 0.7\), \(\gamma_{\{1,2,4\}}^p = 0.5\), \(\gamma_{\{2,3,4\}}^p = 0.3\) and \(\gamma_{\mathfrak u}^p = 0\) otherwise:
```
  --weights projection-dependent:1,2,3:0.7:1,2,4:0.5:2,3,4:0.3
```

Alternatively, projection-dependent weights can be specified in a text file, say weights.txt, with a content similar to the following:

# this is a comment
1,2,3:      0.7
1,2,4:      0.5
2,3,4:      0.3
order 2:    1.0
order 3:    0.1
default:    1.0e-3

This assigns the weights 0.7, 0.5 and 0.3 to projections \(\{1,2,3\}\), \(\{1,2,4\}\) and \(\{2,3,4\}\), respectively, a weight of 1.0 to every projection of order 2, a weight of 0.1 to every projection of order 3 other than the above three for which weights were explicitly assigned, then a weight of \(10^{-3}\) to every other projection. These numerical values are interpreted using the \(p\) power mentioned above. The order of the lines in the file is not important. LatNet Builder must be told where to look for the weights by changing the –weights option to

  --weights file:weights.txt

The first colon-separated token indicates that the weights should be read from a file, and the second one, the (relative or absolute) path to the file to read.

Order-Dependent Weights

In that case, the value of the –weights option consists of three colon-separated tokens: the first is the type of weights (order-dependent weights); the second is the default weight assigned to projection orders for which no weight is explicitly given; the last token is optional and is a comma-separated list of order-dependent weights \(\Gamma_1,\dots,\Gamma_k\) for some \(k\).

Order-dependent weights with \(\Gamma_k^p = 0.3\) for all \(k\):
```
  --weights order-dependent:0.3
```
Order-dependent weights with \(\Gamma_1^p = 1.0\), \(\Gamma_2^p = 0.7\), \(\Gamma_3^p = 0.3\) and \(\Gamma_k^p = 0.1\) for all \(k \geq 4\):
```
  --weights order-dependent:0.1:1.0,0.7,0.3
```

Product Weights

In that case, the value of the –weights option consists of three colon-separated tokens: the first is the type of weights (product weights); the second is the default weight assigned to coordinates for which no weight is explicitly given; the last token is optional and is a comma-separated list of coordinate weights \(\gamma_1,\dots,\gamma_j\) for some \(j\).

Product weights with \(\gamma_j^p = 0.3\) for all \(j\):
```
  --weights product:0.3
```
Product weights with \(\gamma_1^p = 1.0\), \(\gamma_2^p = 0.7\), \(\gamma_3^p = 0.3\) and \(\gamma_j^p = 0.1\) for all \(j \geq 4\):
```
  --weights product:0.1:1.0,0.7,0.3
```

POD Weights

In that case, the value of the –weights option consists of five colon-separated tokens: the first is the type of weights (POD), the next two specify the \(\Gamma_k\)'s as in Order-Dependent Weights, and the last two specify the \(\gamma_j\)'s as in Product Weights.

POD weights with \(\Gamma_1^p = 1.0\), \(\Gamma_2^p = 0.7\), \(\Gamma_3^p = 0.3\) and \(\Gamma_k^p = 0.1\) for all \(k \geq 4\), and with \(\gamma_1^p = 1.0\), \(\gamma_2^p = 0.9\), \(\gamma_3^p = 0.7\) and \(\gamma_j^p = 0.5\) for all \(j \geq 4\):
```
  --weights POD:0.1:1.0,0.7,0.3:0.5:1.0,0.9,0.7
```

Combined Weights

Additionally, one can specify several types of weights as a space separated list of weights descriptions. The weight of a projection will the sum of the specified weights.

Combined weights with \( \gamma_{\mathfrak u}^p = \Gamma_{|\mathfrak u|}^p + \prod_{j \in \mathfrak u} \gamma^p_j \) with the order-dependent weights \( \Gamma \) and the product weights \( \gamma \) defined above in POD Weights :
```
  --weights product:0.1:1.0,0.7,0.3 order-dependent:0.1:1.0,0.7,0.3
```