add nifty jacobian demo

demo_linearization.py

+import nifty8 as ift
+
+from nifty8.sugar import is_linearization
+
+# We want to build a custom model that evaluates the function:
+#               f(x,y) = exp(x) * y + 3
+# and furthermore allows this model to be used in a fit (i.E. the model
+# should also know about its jacobian). To do so, the model should be able to
+# handle inputs being either Fields or Linearizations, with the latter
+# consisting of the input Field and its Jacobian, evaluated at the input Field
+# value. The Jacobian is represented as a LinearOperator which stores how the
+# Jacobian (and its adjoint) can be applied to vectors (Fields) rather then
+# storing the Jacobian as an explicit matrix (which is prohibitively large for
+# bigger applications).
+
+# There are as of now three different ways to achieve this on different levels:
+# 1) The Field level, 2) The Linearization level, 3) The Operator level.
+# The three ways will be constructed in the following.
+
+# Defining domains for the problem:
+# The inputs x,y should live on a scalar domain
+scalar_domain = ift.DomainTuple.scalar_domain()
+# The full input domain (of (x,y) together) should be a `MultiDomain` containing
+# both inputs and assigns the keys 'x' and 'y' to the inputs to identify them.
+full_domain = ift.MultiDomain.make({'x':scalar_domain, 'y':scalar_domain})
+
+
+# 1) The Field level (most general, least fail safe)
+
+# In this most basic form the user defines by hand what should happen in case
+# the input is a Field or a Linearization:
+
+# Define Operator
+class MyCustomFieldModel(ift.Operator):
+    def __init__(self, domain, keys = ('x', 'y')):
+        self._domain = ift.makeDomain(domain)
+        if not isinstance(self._domain, ift.MultiDomain):
+            raise ValueError
+        self._keys = keys
+        for k in self._keys:
+            if k not in self._domain.keys():
+                raise ValueError
+        if len(self._keys) != 2:
+            raise ValueError
+        self._target = ift.makeDomain(domain[self._keys[0]])
+
+    def apply(self, x):
+        self._check_input(x)
+        lin = is_linearization(x) # checks whether x is a linearization or not
+        v = x.val if lin else x # if x is lin the Field values are stored in x.val
+        res = ift.exp(v[self._keys[0]]) * v[self._keys[1]] + 3.
+        if not lin: # in case the input is a Field simply return the result
+            return res
+        # in case x is a Linearization, we also have to provide the Jacobian
+        # and pass it along. (see below)
+        jac = _MyCustomJacobian(v, self._keys)
+        # in this case apply also returns a Linearization.
+        return x.new(res, jac) # creates a new Lin of the correct form (from x).
+
+
+# Definition of the Jacobian
+class _MyCustomJacobian(ift.LinearOperator):
+    def __init__(self, location, keys):
+        self._domain = ift.makeDomain(location.domain)
+        self._keys = keys
+        self._target = ift.makeDomain(self._domain[self._keys[0]])
+        self._capability = self.TIMES | self.ADJOINT_TIMES
+
+        # We can precompute the two derivatives required to build the jacobian
+        self._jy = ift.exp(location[self._keys[0]]) # df/dy = exp(x)
+        self._jx = self._jy * location[self._keys[1]] # df/dx = exp(x) * y
+
+    def apply(self, inp, mode):
+        self._check_input(inp, mode)
+        if mode == self.TIMES:
+            # This part defines the application of the jacobian matrix to an
+            # arbitraty input vector inp
+            return self._jx * inp[self._keys[0]] + self._jy * inp[self._keys[1]]
+        else:
+            # Here the adjoint application is defined, the input in this case is
+            # only a scalar and gets weighted with the entries of the jacobian
+            res = {self._keys[0]: self._jx * inp, self._keys[1]: self._jy * inp}
+            return ift.MultiField.from_dict(res, self._domain)
+
+
+# 2) The Linearization level (very general, more fail safe)
+
+# The behaviour of many basic operations (addition, mutiplication, pointwise 
+# non-linear functions), when facing Fields or Lineariations have already been
+# implemented in nifty. Therefore in many cases one can "do calculations" with
+# Linearizations analogous to Fields and the Jacobians get constructed from the
+# Jacobians of the promitive operations automatically, using the chain rule:
+
+# Define Operator
+class MyCustomLinModel(ift.Operator):
+    def __init__(self, domain, keys = ('x', 'y')):
+        self._domain = ift.makeDomain(domain)
+        if not isinstance(self._domain, ift.MultiDomain):
+            raise ValueError
+        self._keys = keys
+        for k in self._keys:
+            if k not in self._domain.keys():
+                raise ValueError
+        if len(self._keys) != 2:
+            raise ValueError
+        self._target = ift.makeDomain(domain[self._keys[0]])
+
+    def apply(self, x):
+        self._check_input(x)
+        # All operations needed here (input selection, exp, multiplication,
+        # scalar addition) are defined for Fields as well as Linearizations
+        # and therefore the same code can be used in both cases.
+        return ift.exp(x[self._keys[0]]) * x[self._keys[1]] + 3.
+
+
+# 2) The Operator level (a bit less general, most fail safe)
+
+# Similarly to Fields and Linearozations, also the basic operations are
+# implemented for Operators. Given for example two operators op1, op2 we can
+# multiply them together to create a new op3 = op1 * op2. The logic here is:
+# "take the outputs (results) of op1 and op2 and multiply them together". The
+# new operator op3 gets the union of the input domains of op1 and op2 as its
+# domain. The target of op3 is the target of op1 and op2, so only if op1 and op2
+# share the same target space, pointwise multiplication is possible (no
+# automatic broadcasting!). Nifty, however, provides many broadcasting
+# operations the user can invoke to help build more general combinations (see
+# e.g. `ContractionOperator`)
+
+# a placeholder operator that takes 'x' from the input and passes it along.
+x = ift.FieldAdapter(scalar_domain, 'x')
+y = ift.FieldAdapter(scalar_domain, 'y')
+
+# First part of the model. Note that here 'calculations' are performed on
+# operators to create a new operator (i.E. no inputs are provided yet).
+model = ift.exp(x) * y
+# As "+" is already reserved for adding the output of two operators together
+# simple scalar addition has to be performed via a designated operator 'Adder'.
+# Also, on operator level, the combination of operations has to be performed on
+# the same space so the number '3' is cast into a Field on the 'scalar_domain`
+# to add it to the output of model. Finally sequential operator application is
+# given via the matrix multiplication `@`.
+MyCustomModel = ift.Adder(ift.full(scalar_domain, 3.)) @ model
+
+
+# We can verify that the three operators yield the same results on some input.
+inp = ift.from_random(MyCustomModel.domain)
+
+MyFieldModel = MyCustomFieldModel(full_domain)
+MyLinModel = MyCustomLinModel(full_domain)
+
+res1 = MyFieldModel(inp)
+res2 = MyLinModel(inp)
+res3 = MyCustomModel(inp)
+
+ift.extra.assert_allclose(res1, res2)
+ift.extra.assert_allclose(res1, res3)
+
+# We can also build a Linearization and apply the models
+lin = ift.Linearization.make_var(inp)
+
+res1 = MyFieldModel(lin)
+res2 = MyLinModel(lin)
+res3 = MyCustomModel(lin)
+
+# The resulting Field values (accessible via .val) still match.
+ift.extra.assert_allclose(res1.val, res2.val)
+ift.extra.assert_allclose(res1.val, res3.val)
+
+# Looking at the Jacobians evaluated at inp (accessible via .jac), we notice the 
+# differences in implementation of the three approaches.
+print(res1.jac)
+print("------")
+print(res2.jac)
+print("------")
+print(res3.jac)
+
+# Applying these jacobians (and their adjoint) to random inputs, however,
+# gives the same results as they all apply the same matrix mathematically.
+jac_inp = ift.from_random(res1.jac.domain)
+
+ift.extra.assert_allclose(res1.jac(jac_inp), res2.jac(jac_inp))
+ift.extra.assert_allclose(res1.jac(jac_inp), res3.jac(jac_inp))
+
+# same for the adjoint application
+jac_inp = ift.from_random(res1.jac.target)
+ift.extra.assert_allclose(res1.jac.adjoint(jac_inp), res2.jac.adjoint(jac_inp))
+ift.extra.assert_allclose(res1.jac.adjoint(jac_inp), res3.jac.adjoint(jac_inp))
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