agd.Metrics.isotropic

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 1# Copyright 2020 Jean-Marie Mirebeau, University Paris-Sud, CNRS, University Paris-Saclay
 2# Distributed WITHOUT ANY WARRANTY. Licensed under the Apache License, Version 2.0, see http://www.apache.org/licenses/LICENSE-2.0
 3
 4from .base import Base
 5from .. import AutomaticDifferentiation as ad
 6import numpy as np
 7
 8class Isotropic(Base):
 9	r"""
10	An Isotropic norm takes the form 
11	$$
12	F(x) = cost * \sqrt{< x,x>},
13	$$
14	where cost is a given positive scalar.
15
16	Member fields and __init__ arguments : 
17	- cost : an array of arbirary shape (n1,..,nk).
18	- vdim (optional) : the ambient space dimension
19	"""
20
21	def __init__(self,cost,vdim=None):
22		self.cost = ad.asarray(cost)
23		self._vdim = vdim
24
25	@classmethod
26	def from_speed(cls,speed,vdim=None):
27		"""
28		Produces a metric whose cost equals 1/speed.
29		"""
30		return cls(1./speed,vdim)
31
32	def dual(self):
33		other = self.from_speed(self.cost)
34		if hasattr(self,'_vdim'): other._vdim = self._vdim
35		return other
36
37	def norm(self,v):
38		return self.cost*ad.Optimization.norm(v,ord=2,axis=0)
39
40	def gradient(self,v):
41		return self.cost*v/ad.Optimization.norm(v,ord=2,axis=0)
42
43	def is_definite(self):
44		return self.cost>0.
45
46	def anisotropy(self):
47		return 1.
48
49	def cost_bound(self):
50		return self.cost
51
52	@property
53	def vdim(self): 
54		if self._vdim is not None: return self._vdim
55		if self.cost.ndim>0: return self.cost.ndim
56		raise ValueError("Could not determine dimension of isotropic metric")
57
58	@vdim.setter
59	def vdim(self,vdim):
60		if self.cost.ndim>0:
61			assert(self.cost.ndim==vdim)
62		else:
63			self._vdim = vdim
64
65	@property
66	def shape(self): return self.cost.shape
67	
68	def rotate(self,a): return self
69	def with_cost(self,cost): return Isotropic(cost*self.cost)
70	def with_costs(self,costs):
71		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
72			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
73		return self.with_cost(costs[0])
74
75	def flatten(self):      return self.cost
76	@classmethod
77	def expand(cls,arr):    return cls(arr)
78
79	def to_HFM(self):       return self.cost
80	@classmethod
81	def from_HFM(cls,arr):  return cls(arr)
82
83	def model_HFM(self):
84		return "Isotropic"+str(self.vdim)
85
86	@classmethod
87	def from_cast(cls,metric):
88		if isinstance(metric,cls): return metric
89		raise ValueError("Isotropic.from_cast error : cannot cast an isotropic metric from ",type(metric))
90
91	def __iter__(self):
92		yield self.cost
93
94	def make_proj_dual(self):
95		"""Returns the projection operator onto the unit ball of the dual metric"""
96		a=1./self.cost
97		norm = np.linalg.norm
98		return lambda x : x / np.maximum(1.,a*norm(x,axis=0))

class Isotropic(agd.Metrics.base.Base): View Source

 9class Isotropic(Base):
10	r"""
11	An Isotropic norm takes the form 
12	$$
13	F(x) = cost * \sqrt{< x,x>},
14	$$
15	where cost is a given positive scalar.
16
17	Member fields and __init__ arguments : 
18	- cost : an array of arbirary shape (n1,..,nk).
19	- vdim (optional) : the ambient space dimension
20	"""
21
22	def __init__(self,cost,vdim=None):
23		self.cost = ad.asarray(cost)
24		self._vdim = vdim
25
26	@classmethod
27	def from_speed(cls,speed,vdim=None):
28		"""
29		Produces a metric whose cost equals 1/speed.
30		"""
31		return cls(1./speed,vdim)
32
33	def dual(self):
34		other = self.from_speed(self.cost)
35		if hasattr(self,'_vdim'): other._vdim = self._vdim
36		return other
37
38	def norm(self,v):
39		return self.cost*ad.Optimization.norm(v,ord=2,axis=0)
40
41	def gradient(self,v):
42		return self.cost*v/ad.Optimization.norm(v,ord=2,axis=0)
43
44	def is_definite(self):
45		return self.cost>0.
46
47	def anisotropy(self):
48		return 1.
49
50	def cost_bound(self):
51		return self.cost
52
53	@property
54	def vdim(self): 
55		if self._vdim is not None: return self._vdim
56		if self.cost.ndim>0: return self.cost.ndim
57		raise ValueError("Could not determine dimension of isotropic metric")
58
59	@vdim.setter
60	def vdim(self,vdim):
61		if self.cost.ndim>0:
62			assert(self.cost.ndim==vdim)
63		else:
64			self._vdim = vdim
65
66	@property
67	def shape(self): return self.cost.shape
68	
69	def rotate(self,a): return self
70	def with_cost(self,cost): return Isotropic(cost*self.cost)
71	def with_costs(self,costs):
72		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
73			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
74		return self.with_cost(costs[0])
75
76	def flatten(self):      return self.cost
77	@classmethod
78	def expand(cls,arr):    return cls(arr)
79
80	def to_HFM(self):       return self.cost
81	@classmethod
82	def from_HFM(cls,arr):  return cls(arr)
83
84	def model_HFM(self):
85		return "Isotropic"+str(self.vdim)
86
87	@classmethod
88	def from_cast(cls,metric):
89		if isinstance(metric,cls): return metric
90		raise ValueError("Isotropic.from_cast error : cannot cast an isotropic metric from ",type(metric))
91
92	def __iter__(self):
93		yield self.cost
94
95	def make_proj_dual(self):
96		"""Returns the projection operator onto the unit ball of the dual metric"""
97		a=1./self.cost
98		norm = np.linalg.norm
99		return lambda x : x / np.maximum(1.,a*norm(x,axis=0))

An Isotropic norm takes the form $$ F(x) = cost * \sqrt{< x,x>}, $$ where cost is a given positive scalar.

Member fields and __init__ arguments :

cost : an array of arbirary shape (n1,..,nk).
vdim (optional) : the ambient space dimension

Isotropic(cost, vdim=None) View Source

22	def __init__(self,cost,vdim=None):
23		self.cost = ad.asarray(cost)
24		self._vdim = vdim

cost

@classmethod

def from_speed(cls, speed, vdim=None): View Source

26	@classmethod
27	def from_speed(cls,speed,vdim=None):
28		"""
29		Produces a metric whose cost equals 1/speed.
30		"""
31		return cls(1./speed,vdim)

Produces a metric whose cost equals 1/speed.

def dual(self): View Source

33	def dual(self):
34		other = self.from_speed(self.cost)
35		if hasattr(self,'_vdim'): other._vdim = self._vdim
36		return other

Dual norm, mathematically defined by $N^*(x) = max\{ < x, y> ; N(y)\leq 1 \}$

def norm(self, v): View Source

38	def norm(self,v):
39		return self.cost*ad.Optimization.norm(v,ord=2,axis=0)

Norm or quasi-norm defined by the class, often denoted $N$ in mathematical formulas. Unless incorrect data is provided, this member function obeys, for all vectors $u,v\in R^d$ and all $\alpha \geq 0$

$N(u+v) \leq N(u)+N(v)$
$N(\alpha u) = \alpha N(u)$
$N(u)\geq 0$ with equality iff $u=0$.

Broadcasting will occur depending on the shape of $v$ and of the class data.

def gradient(self, v): View Source

41	def gradient(self,v):
42		return self.cost*v/ad.Optimization.norm(v,ord=2,axis=0)

Gradient of the norm defined by the class.

def is_definite(self): View Source

44	def is_definite(self):
45		return self.cost>0.

Attempts to check wether the data defines a mathematically valid norm.

def anisotropy(self): View Source

47	def anisotropy(self):
48		return 1.

Anisotropy ratio of the norm denoted $N$. Defined as $$ \max_{|u| = |v| = 1} \frac {N(u)}{N(v)}. $$

def cost_bound(self): View Source

50	def cost_bound(self):
51		return self.cost

Upper bound on $N(u)$, for any unit vector $u$, where $N$ is the norm defined by the class.

vdim View Source

53	@property
54	def vdim(self): 
55		if self._vdim is not None: return self._vdim
56		if self.cost.ndim>0: return self.cost.ndim
57		raise ValueError("Could not determine dimension of isotropic metric")

The ambient vector space dimension, often denoted $d$ in mathematical formulas.

shape View Source

66	@property
67	def shape(self): return self.cost.shape

Dimensions of the underlying domain. Expected to be the empty tuple, or a tuple of length vdim.

def rotate(self, a): View Source

69	def rotate(self,a): return self

Rotation of the norm, by a given rotation matrix. The new unit ball is the direct image of the previous one.

def with_cost(self, cost): View Source

70	def with_cost(self,cost): return Isotropic(cost*self.cost)

Produces a norm $N'$ obeying $N'(x) = N(cost*x)$.

def with_costs(self, costs): View Source

71	def with_costs(self,costs):
72		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
73			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
74		return self.with_cost(costs[0])

Produces a norm $N'$ defined by $$ N'(x) = N(costs * x) $$ where the multiplication is elementwise.

def flatten(self): View Source

76	def flatten(self):      return self.cost

Flattens and concatenate the member fields into a single array.

@classmethod

def expand(cls, arr): View Source

77	@classmethod
78	def expand(cls,arr):    return cls(arr)

Inverse of the flatten member function. Turns a suitable array into a metric.

def to_HFM(self): View Source

80	def to_HFM(self):       return self.cost

Formats a metric for the HFM library. This may include flattening some symmetric matrices, concatenating with vector fields, and moving the first axis last.

@classmethod

def from_HFM(cls, arr): View Source

81	@classmethod
82	def from_HFM(cls,arr):  return cls(arr)

Inverse of the to_HFM member function. Turns a suitable array into a metric.

def model_HFM(self): View Source

84	def model_HFM(self):
85		return "Isotropic"+str(self.vdim)

The name of the 'model' for parameter, as input to the HFM library.

@classmethod

def from_cast(cls, metric): View Source

87	@classmethod
88	def from_cast(cls,metric):
89		if isinstance(metric,cls): return metric
90		raise ValueError("Isotropic.from_cast error : cannot cast an isotropic metric from ",type(metric))

Produces a metric by casting another metric of a compatible type.

def make_proj_dual(self): View Source

95	def make_proj_dual(self):
96		"""Returns the projection operator onto the unit ball of the dual metric"""
97		a=1./self.cost
98		norm = np.linalg.norm
99		return lambda x : x / np.maximum(1.,a*norm(x,axis=0))

Returns the projection operator onto the unit ball of the dual metric

Inherited Members

agd.Metrics.base.Base: disassociate; anisotropy_bound; cos_asym; cos; angle; inv_transform; transform; rotate_by; with_speeds; with_speed; from_generator; array_float_caster; norm2; gradient2; set_interpolation; at; make_proj; make_prox