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PersLay parameters
------------------
In the following description of PersLay parameters, each parameter, or dictionary key, that contains `_init` in its name is optimized and learned by PersLay during training. If you do not want to optimize the vectorization, set the keys **train_vect** and **train_weight** to False.
* The following keys are mandatory:
+ layer
Either "PermutationEquivariant", "Image", "Landscape", "BettiCurve", "Entropy", "Exponential", "Rational" or "RationalHat". Type of the PersLay layer. "Image" is for `persistence images <https://arxiv.org/abs/1507.06217>`_, "Landscape" is for `persistence landscapes <http://www.jmlr.org/papers/volume16/bubenik15a/bubenik15a.pdf>`_, "Exponential", "Rational" and "RationalHat" are for `structure elements <http://jmlr.org/beta/papers/v20/18-358.html>`_, "PermutationEquivariant" is for the original DeepSet layer, defined in `this article <https://arxiv.org/abs/1703.06114>`_, "BettiCurve" is for `Betti curves <https://www.jstage.jst.go.jp/article/tjsai/32/3/32_D-G72/_pdf>`_ and "Entropy" is for `entropy <https://arxiv.org/abs/1803.08304>`_.
+ perm_op
Either "sum", "mean", "max", "topk". Permutation invariant operation.
+ keep
Number of top values to keep. Used only if **perm_op** is "topk".
+ pweight
Either "power", "grid", "gmix" or None. Weight function to be applied on persistence diagram points. If "power", this function is a (trainable) coefficient times the distances to the diagonal of the points to a certain power. If "grid", this function is piecewise-constant and defined with pixel values of a grid. If "gmix", this function is defined as a mixture of Gaussians. If None, no weighting is applied.
+ final_model
A Tensorflow / Keras model used to postprocess the persistence diagrams in each channel. Use "identity" if you don't want to postprocess.
* Depending on what **pweight** is, the following additional keys are requested:
+ if **pweight** is "power":
- pweight_init
Initializer of the coefficient of the power weight function. It can be either a single value, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- pweight_power
Integer used for exponentiating the distances to the diagonal of the persistence diagram points.
+ if **pweight** is "grid":
- pweight_size
Grid size of the grid weight function. It is a tuple of integer values, such as (10,10).
- pweight_bnds
Grid boundaries of the grid weight function. It is a tuple containing two tuples, each containing the minimum and maximum values of each axis of the plane. Example: ((-0.01, 1.01), (-0.01, 1.01)).
- pweight_init
Initializer for the pixel values of the grid weight function. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
+ if **pweight** is "gmix":
- pweight_num
Number of Gaussian functions of the mixture of Gaussians weight function.
- pweight_init
Initializer of the means and variances of the mixture of Gaussians weight function. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
* Depending on what **layer** is, the following additional keys are requested:
+ if **layer** is "PermutationEquivariant":
- lpeq
Sequence of permutation equivariant operations, as defined in [the DeepSet article](). It is a list of tuples of the form (*dim*, *operation*). Each tuple defines a permutation equivariant function of dimension *dim* and second permutation operation *operation* (string, either "max", "min", "sum" or None). Second permutation operation is optional and is not applied if *operation* is set to None. Example: [(150, "max"), (75, None)].
- lweight_init
Initializer for the weight matrices of the permutation equivariant operations. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- lbias_init
Initializer for the biases of the permutation equivariant operations. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- lgamma_init
Initializer for the Gamma matrices of the permutation equivariant operations. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
+ if **layer** is "Image":
- image_size
Persistence image size. It is a tuple of integer values, such as (10,10).
- image_bnds
Persistence image boundaries. It is a tuple containing two tuples, each containing the minimum and maximum values of each axis of the plane. Example: ((-0.01, 1.01), (-0.01, 1.01)).
- lvariance_init
Initializer for the bandwidths of the Gaussian functions centered on the persistence image pixels. It can be either a single value, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 3.).
+ if **layer** is "Landscape":
- lsample_num
Number of samples of the diagonal that will be evaluated on the persistence landscapes.
- lsample_init
Initializer of the samples of the diagonal. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
+ if **layer** is "BettiCurve":
- lsample_num
Number of samples of the diagonal that will be evaluated on the Betti curves.
- lsample_init
Initializer of the samples of the diagonal. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- theta
Sigmoid parameter used for approximating the piecewise constant functions associated to the persistence diagram points.
+ if **layer** is "Entropy":
- lsample_num
Number of samples on the diagonal that will be evaluated on the persistence entropies.
- lsample_init
Initializer of the samples of the diagonal. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- theta
Sigmoid parameter used for approximating the piecewise constant functions associated to the persistence diagram points.
+ if **layer** is "Exponential":
- lnum
Number of exponential structure elements that will be evaluated on the persistence diagram points.
- lmean_init
Initializer of the means of the exponential structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- lvariance_init
Initializer of the bandwidths of the exponential structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(3., 3.).
+ if **layer** is "Rational":
- lnum
Number of rational structure elements that will be evaluated on the persistence diagram points.
- lmean_init
Initializer of the means of the rational structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- lvariance_init
Initializer of the bandwidths of the rational structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(3., 3.).
- lalpha_init
Initializer of the exponents of the rational structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(3., 3.).
+ if **layer** is "RationalHat":
- lnum
Number of rational hat structure elements that will be evaluated on the persistence diagram points.
- lmean_init
Initializer of the means of the rational hat structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(0., 1.).
- lr_init
Initializer of the threshold of the rational hat structure elements. It can be either a numpy array of values, or a random initializer from tensorflow, such as tensorflow.random_uniform_initializer(3., 3.).
- q
Norm parameter.
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