"""
This file implements the Mean & Covariance RBM discussed in 

    Ranzato, M. and Hinton, G. E. (2010)
    Modeling pixel means and covariances using factored third-order Boltzmann machines.
    IEEE Conference on Computer Vision and Pattern Recognition.

and performs one of the experiments on CIFAR-10 discussed in that paper.  There are some minor
discrepancies between the paper and the accompanying code (train_mcRBM.py), and the
accompanying code has been taken to be correct in those cases because I couldn't get things to
work otherwise.


Math
====

Energy of "covariance RBM"

    E = -0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i C_{if} v_i )^2
      = -0.5 \sum_f (\sum_k P_{fk} h_k) ( \sum_i C_{if} v_i )^2
                    "vector element f"           "vector element f"

In some parts of the paper, the P matrix is chosen to be a diagonal matrix with non-positive
diagonal entries, so it is helpful to see this as a simpler equation:

    E =  \sum_f h_f ( \sum_i C_{if} v_i )^2



Version in paper
----------------

Full Energy of the Mean and Covariance RBM, with 
:math:`h_k = h_k^{(c)}`,
:math:`g_j = h_j^{(m)}`,
:math:`b_k = b_k^{(c)}`,
:math:`c_j = b_j^{(m)}`,
:math:`U_{if} = C_{if}`,

    E (v, h, g) =
        - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i (U_{if} v_i) / |U_{.f}|*|v| )^2 
        - \sum_k b_k h_k
        + 0.5 \sum_i v_i^2
        - \sum_j \sum_i W_{ij} g_j v_i
        - \sum_j c_j g_j

For the energy function to correspond to a probability distribution, P must be non-positive.  P
is initialized to be a diagonal, and in our experience it can be left as such because even in
the paper it has a very low learning rate, and is only allowed to be updated after the filters
in U are learned (in effect).

Version in published train_mcRBM code
-------------------------------------

The train_mcRBM file implements learning in a similar but technically different Energy function:

    E (v, h, g) =
        - 0.5 \sum_f \sum_k P_{fk} h_k (\sum_i U_{if} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2 
        - \sum_k b_k h_k
        + 0.5 \sum_i v_i^2
        - \sum_j \sum_i W_{ij} g_j v_i
        - \sum_j c_j g_j

There are two differences with respect to the paper:

    - 'v' is not normalized by its length, but rather it is normalized to have length close to
      the square root of the number of its components.  The variable called 'small' that
      "avoids division by zero" is orders larger than machine precision, and is on the order of
      the normalized sum-of-squares, so I've included it in the Energy function.

    - 'U' is also not normalized by its length.  U is initialized to have columns that are
      shorter than unit-length (approximately 0.2 with the 105 principle components in the
      train_mcRBM data).  During training, the columns of U are constrained manually to have
      equal lengths (see the use of normVF), but Euclidean norm is allowed to change.  During
      learning it quickly converges towards 1 and then exceeds 1.  It does not seem like this
      column-wise normalization of U is justified by maximum-likelihood, I have no intuition
      for why it is used.


Version in this code
--------------------

This file implements the same algorithm as the train_mcRBM code, except that the P matrix is
omitted for clarity, and replaced analytically with a negative identity matrix.

    E (v, h, g) =
        + 0.5 \sum_k h_k (\sum_i U_{ik} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2 
        - \sum_k b_k h_k
        + 0.5 \sum_i v_i^2
        - \sum_j \sum_i W_{ij} g_j v_i
        - \sum_j c_j g_j

      

Conventions in this file
========================

This file contains some global functions, as well as a class (MeanCovRBM) that makes using them a little
more convenient.


Global functions like `free_energy` work on an mcRBM as parametrized in a particular way.
Suppose we have 
I input dimensions, 
F squared filters, 
J mean variables, and 
K covariance variables.
The mcRBM is parametrized by 5 variables:

 - `U`, a matrix whose rows are visible covariance directions (I x F)
 - `W`, a matrix whose rows are visible mean directions (I x J)
 - `b`, a vector of hidden covariance biases (K)
 - `c`, a vector of hidden mean biases  (J)

Matrices are generally layed out and accessed according to a C-order convention.

"""

#
# WORKING NOTES
# THIS DERIVATION IS BASED ON THE ** PAPER ** ENERGY FUNCTION
# NOT THE ENERGY FUNCTION IN THE CODE!!!
#
# Free energy is the marginal energy of visible units
# Recall: 
#   Q(x) = exp(-E(x))/Z ==> -log(Q(x)) - log(Z) = E(x)
#
#
#   E (v, h, g) =
#       - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / |U_{*f}|^2 |v|^2
#       - \sum_k b_k h_k
#       + 0.5 \sum_i v_i^2
#       - \sum_j \sum_i W_{ij} g_j v_i
#       - \sum_j c_j g_j
#       - \sum_i a_i v_i
#
#
# Derivation, in which partition functions are ignored.
#
# E(v) = -\log(Q(v))
#  = -\log( \sum_{h,g} Q(v,h,g))
#  = -\log( \sum_{h,g} exp(-E(v,h,g)))
#  = -\log( \sum_{h,g} exp(-
#       - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|)
#       - \sum_k b_k h_k
#       + 0.5 \sum_i v_i^2
#       - \sum_j \sum_i W_{ij} g_j v_i
#       - \sum_j c_j g_j 
#       - \sum_i a_i v_i ))
#
# Get rid of double negs  in exp
#  = -\log(  \sum_{h} exp(
#       + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|)
#       + \sum_k b_k h_k
#       - 0.5 \sum_i v_i^2
#       ) * \sum_{g} exp(
#       + \sum_j \sum_i W_{ij} g_j v_i
#       + \sum_j c_j g_j))
#    - \sum_i a_i v_i 
#
# Break up log
#  = -\log(  \sum_{h} exp(
#       + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|)
#       + \sum_k b_k h_k
#       ))
#    -\log( \sum_{g} exp(
#       + \sum_j \sum_i W_{ij} g_j v_i
#       + \sum_j c_j g_j )))
#    + 0.5 \sum_i v_i^2
#    - \sum_i a_i v_i 
#
# Use domain h is binary to turn log(sum(exp(sum...))) into sum(log(..
#  = -\log(\sum_{h} exp(
#       + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|* |v|)
#       + \sum_k b_k h_k
#       ))
#    - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
#    + 0.5 \sum_i v_i^2
#    - \sum_i a_i v_i 
#
#  = - \sum_{k} \log(1 + exp(b_k + 0.5 \sum_f P_{fk}( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|)))
#    - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
#    + 0.5 \sum_i v_i^2
#    - \sum_i a_i v_i 
#
# For negative-one-diagonal P this gives:
#
#  = - \sum_{k} \log(1 + exp(b_k - 0.5 \sum_i (U_{ik} v_i )^2 / (|U_{*k}|*|v|)))
#    - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
#    + 0.5 \sum_i v_i^2
#    - \sum_i a_i v_i 

import sys
import logging
import numpy as np
import numpy

import theano
from theano import function, shared, dot
from theano import tensor as TT
floatX = theano.config.floatX

import pylearn
from pylearn.sampling.hmc import HMC_sampler
from pylearn.io import image_tiling
from pylearn.gd.sgd import sgd_updates

#TODO: This should be in the datasets folder
import pylearn.datasets.config
import pylearn.dataset_ops.image_patches
from pylearn.dataset_ops.protocol import TensorFnDataset
from pylearn.dataset_ops.memo import memo
import pylearn
import scipy.io
import os


#TODO: This should be in the nnet part of the library
def hidden_cov_units_preactivation_given_v(rbm, v, small=0.5):
    """Return argument to the sigmoid that would give mean of covariance hid units

    See the math at the top of this file for what 'adjusted' means.

    return b - 0.5 * dot(adjusted(v), U)**2
    """
    (U,W,a,b,c) = rbm
    unit_v = v / (TT.sqrt(TT.mean(v**2, axis=1)+small)).dimshuffle(0,'x') # adjust row norm
    return b - 0.5 * dot(unit_v, U)**2

def free_energy_terms_given_v(rbm, v):
    """Returns theano expression for the terms that are added to form the free energy of
    visible vector `v` in an mcRBM.

     1.  Free energy related to covariance hiddens
     2.  Free energy related to mean hiddens
     3.  Free energy related to L2-Norm of `v`
     4.  Free energy related to projection of `v` onto biases `a`
    """
    U, W, a, b, c = rbm
    t0 = -TT.sum(TT.nnet.softplus(hidden_cov_units_preactivation_given_v(rbm, v)),axis=1)
    t1 = -TT.sum(TT.nnet.softplus(c + dot(v,W)), axis=1)
    t2 =  0.5 * TT.sum(v**2, axis=1)
    t3 = -TT.dot(v, a)
    return [t0, t1, t2, t3]

def free_energy_given_v(rbm, v):
    """Returns theano expression for free energy of visible vector `v` in an mcRBM
    """
    return sum(free_energy_terms_given_v(rbm,v))

def contrastive_gradient(rbm, pos_v, neg_v, U_l1_penalty=0, W_l1_penalty=0):
    """Return a list of gradient expressions for the rbm parameters

    :param pos_v: positive-phase sample of visible units
    :param neg_v: negative-phase sample of visible units
    :param U_l1_penalty: a scalar-valued multiplier on the L1 penalty on U
    :param W_l1_penalty: a scalar-valued multiplier on the L1 penalty on W
    """
    U, W, a, b, c = rbm
    pos_FE = free_energy_given_v(rbm, pos_v)
    neg_FE = free_energy_given_v(rbm, neg_v)
    c0 = (pos_FE - neg_FE).sum()
    c1 = abs(U).sum()*U_l1_penalty 
    c2 = abs(W).sum()*W_l1_penalty 
    cost = c0 + c1 + c2
    rval = theano.tensor.grad(cost, list(rbm))
    return rval

def expected_h_g_given_v(rbm, v):
    """Returns tuple (`h`, `g`) of theano expression conditional expectations in an mcRBM.

    `h` is the conditional on the covariance units.
    `g` is the conditional on the mean units.
    
    """
    (U, W, a, b, c) = rbm
    h = TT.nnet.sigmoid(hidden_cov_units_preactivation_given_v(rbm, v))
    g = nnet.sigmoid(c + dot(v,W))
    return (h, g)

def n_visible_units(rbm):
    """Return the number of visible units of this RBM

    For an RBM made from shared variables, this will return an integer,
    for a purely symbolic RBM this will return a theano expression.
    
    """
    W = rbm[1]
    try:
        return W.value.shape[0]
    except AttributeError:
        return W.shape[0]


def sampler(rbm, n_particles, n_visible=None, rng=7823748):
    """Return an `HMC_sampler` that will draw samples from the distribution over visible
    units specified by this RBM.

    :param n_particles: this many parallel chains will be simulated.
    :param rng: seed or numpy RandomState object to initialize particles, and to drive the simulation.
    """
    if not hasattr(rng, 'randn'):
        rng = np.random.RandomState(rng)
    if n_visible is None:
        n_visible = n_visible_units(rbm)
    rval = HMC_sampler(
        positions = [shared(
            rng.randn(
                n_particles,
                n_visible).astype(floatX),
            name='particles')],
        energy_fn = lambda p : free_energy_given_v(rbm, p[0]),
        seed=int(rng.randint(2**30)))
    return rval

class MeanCovRBM(object):
    """Container for mcRBM parameters

    It provides parameter lookup by name, as well as a heuristic for initializing the
    parameters for effective learning.

    Attributes:

      - U - the covariance filters (theano shared variable)
      - W - the mean filters (theano shared variable)
      - a - the visible bias (theano shared variable)
      - b - the covariance bias (theano shared variable)
      - c - the mean bias (theano shared variable)
    
    """

    params = property(lambda s: [s.U, s.W, s.a, s.b, s.c])

    n_visible = property(lambda s: s.W.value.shape[0])

    def __init__(self, U, W, a, b, c):
        self.__dict__.update(locals())
        del self.__dict__['self']

    def __getitem__(self, idx):
        # support unpacking of this container as if it were a tuple
        return self.params[idx]

    @classmethod
    def new_from_dims(cls, n_I, n_K, n_J, rng = 8923402190):
        """
        Return a MeanCovRBM instance with randomly-initialized parameters.

        :param n_I: input dimensionality
        :param n_K: number of covariance hidden units
        :param n_J: number of mean filters (linear)
        :param rng: seed or numpy RandomState object to initialize params
        """
        if not hasattr(rng, 'randn'):
            rng = np.random.RandomState(rng)

        def shrd(X,name):
            return shared(X.astype(floatX), name=name)

        # initialization taken from train_mcRBM.py
        return cls(
                U = shrd(0.02 * rng.randn(n_I, n_K),'U'),
                W = shrd(0.05 * rng.randn(n_I, n_J),'W'),
                a = shrd(np.ones(n_I)*(0),'a'),
                b = shrd(np.ones(n_K)*2,'b'),
                c = shrd(np.ones(n_J)*(-2),'c'))

    def __getstate__(self):
        # unpack shared containers, which may have references to Theano stuff and are not a
        # long-term stable data type.
        return dict(
                U = self.U.value,
                W = self.W.value,
                b = self.b.value,
                c = self.c.value)

    def __setstate__(self, dct):
        d = dict(dct)
        for key in ['U', 'W', 'a', 'b', 'c']:
            d[key] = shared(d[key], name=key)
        self.__init__(**d)

if __name__ == '__main__':

    dataset='MAR'
    if dataset == 'MAR':
        n_vis=105
        n_patches=10240
    else:
        R,C= 16,16 # the size of image patches
        n_vis=R*C
        n_patches=100000

    n_train_iters=5000

    n_burnin_steps=10000

    l1_penalty=1e-3
    no_l1_epochs = 10
    effective_l1_penalty=0.0

    epoch_size=n_patches
    batchsize = 128
    lr = 0.075 / batchsize
    s_lr = TT.scalar()
    s_l1_penalty=TT.scalar()
    n_K=256
    n_J=100

    rbm = MeanCovRBM.new_from_dims(n_I=n_vis, n_K=n_K, n_J=n_J) 

    smplr = sampler(rbm, n_particles=batchsize)

    def l2(X):
        return numpy.sqrt((X**2).sum())
    if dataset == 'MAR':
        tile = pylearn.dataset_ops.image_patches.save_filters_of_ranzato_hinton_2010
    else:
        def tile(X, fname):
            _img = image_tiling.tile_raster_images(X,
                    img_shape=(R,C),
                    min_dynamic_range=1e-2)
            image_tiling.save_tiled_raster_images(_img, fname)

    batch_idx = TT.iscalar()

    if dataset == 'MAR':
        train_batch = pylearn.dataset_ops.image_patches.ranzato_hinton_2010_op(batch_idx * batchsize + np.arange(batchsize))
    else:
        train_batch = pylearn.dataset_ops.image_patches.image_patches(
                s_idx = (batch_idx * batchsize + np.arange(batchsize)),
                dims = (n_patches,R,C),
                center=True,
                unitvar=True,
                dtype=floatX,
                rasterized=True)

    imgs_fn = function([batch_idx], outputs=train_batch)

    grads = contrastive_gradient(rbm,
            pos_v=train_batch, 
            neg_v=smplr.positions[0],
            U_l1_penalty=s_l1_penalty,
            W_l1_penalty=s_l1_penalty)
    sgd_ups = sgd_updates(
                rbm.params,
                grads,
                stepsizes=[2*s_lr, .2*s_lr, .02*s_lr, .1*s_lr, .02*s_lr ])
    learn_fn = function([batch_idx, s_lr, s_l1_penalty], 
            outputs=[ 
                grads[0].norm(2),
                (sgd_ups[0][1] - sgd_ups[0][0]).norm(2),
                (sgd_ups[1][1] - sgd_ups[1][0]).norm(2),
                ],
            updates = sgd_ups)

    print "Learning..."
    normVF=1
    last_epoch = -1
    for jj in xrange(n_train_iters):
        epoch = jj*batchsize / epoch_size

        print_jj = epoch != last_epoch
        last_epoch = epoch

        if epoch > 10:
            break

        if print_jj:
            tile(imgs_fn(jj), "imgs_%06i.png"%jj)
            tile(smplr.positions[0].value, "sample_%06i.png"%jj)
            tile(rbm.U.value.T, "U_%06i.png"%jj)
            tile(rbm.W.value.T, "W_%06i.png"%jj)

            print 'saving samples', jj, 'epoch', jj/(epoch_size/batchsize)

            print 'l2(U)', l2(rbm.U.value),
            print 'l2(W)', l2(rbm.W.value)

            print 'U min max', rbm.U.value.min(), rbm.U.value.max(),
            print 'W min max', rbm.W.value.min(), rbm.W.value.max(),
            print 'a min max', rbm.a.value.min(), rbm.a.value.max(),
            print 'b min max', rbm.b.value.min(), rbm.b.value.max(),
            print 'c min max', rbm.c.value.min(), rbm.c.value.max()

            print 'parts min', smplr.positions[0].value.min(), 
            print 'max',smplr.positions[0].value.max(),
            print 'HMC step', smplr.stepsize,
            print 'arate', smplr.avg_acceptance_rate

        smplr.simulate()

        l2_of_Ugrad = learn_fn(jj, 
                lr/max(1, jj/(20*epoch_size/batchsize)),
                effective_l1_penalty)

        if print_jj:
            print 'l2(U_grad)', float(l2_of_Ugrad[0]),
            print 'l2(U_inc)', float(l2_of_Ugrad[1]),
            print 'l2(W_inc)', float(l2_of_Ugrad[2]),
            #print 'FE+', float(l2_of_Ugrad[2]),
            #print 'FE+[0]', float(l2_of_Ugrad[3]),
            #print 'FE+[1]', float(l2_of_Ugrad[4]),
            #print 'FE+[2]', float(l2_of_Ugrad[5]),
            #print 'FE+[3]', float(l2_of_Ugrad[6])

        if jj == no_l1_epochs * epoch_size/batchsize:
            print "Activating L1 weight decay"
            effective_l1_penalty = 1e-3

        # weird normalization technique...
        # It constrains all the columns of the matrix to have the same length
        # But the matrix itself is re-scaled to have an arbitrary abslute size.
        U = rbm.U.value
        U_norms = np.sqrt((U*U).sum(axis=0))
        assert len(U_norms) == n_K
        normVF = .95 * normVF + .05 * np.mean(U_norms)
        rbm.U.value = rbm.U.value * normVF/U_norms