__docformat__ = "restructuredtext en" from mdp import numx, numx_linalg, Cumulator, TrainingException, MDPWarning from mdp.utils import mult, nongeneral_svd, svd, sqrtm, symeig import warnings as _warnings # some useful functions sqrt = numx.sqrt # search XXX for locations where future work is needed ######################################################### # Locally Linear Embedding ######################################################### class LLENode(Cumulator): """Perform a Locally Linear Embedding analysis on the data. **Internal variables of interest** ``self.training_projection`` The LLE projection of the training data (defined when training finishes). ``self.desired_variance`` variance limit used to compute intrinsic dimensionality. Based on the algorithm outlined in *An Introduction to Locally Linear Embedding* by L. Saul and S. Roweis, using improvements suggested in *Locally Linear Embedding for Classification* by D. deRidder and R.P.W. Duin. References: Roweis, S. and Saul, L., Nonlinear dimensionality reduction by locally linear embedding, Science 290 (5500), pp. 2323-2326, 2000. Original code contributed by: Jake VanderPlas, University of Washington, """ def __init__(self, k, r=0.001, svd=False, verbose=False, input_dim=None, output_dim=None, dtype=None): """ :Arguments: k number of nearest neighbors to use r regularization constant; if ``None``, ``r`` is automatically computed using the method presented in deRidder and Duin; this method involves solving an eigenvalue problem for every data point, and can slow down the algorithm If specified, it multiplies the trace of the local covariance matrix of the distances, as in Saul & Roweis (faster) svd if true, use SVD to compute the projection matrix; SVD is slower but more stable verbose if true, displays information about the progress of the algorithm output_dim number of dimensions to output or a float between 0.0 and 1.0. In the latter case, ``output_dim`` specifies the desired fraction of variance to be explained, and the final number of output dimensions is known at the end of training (e.g., for ``output_dim=0.95`` the algorithm will keep as many dimensions as necessary in order to explain 95% of the input variance) """ if isinstance(output_dim, float) and output_dim <= 1: self.desired_variance = output_dim output_dim = None else: self.desired_variance = None super(LLENode, self).__init__(input_dim, output_dim, dtype) self.k = k self.r = r self.svd = svd self.verbose = verbose def _stop_training(self): Cumulator._stop_training(self) if self.verbose: msg = ('training LLE on %i points' ' in %i dimensions...' % (self.data.shape[0], self.data.shape[1])) print msg # some useful quantities M = self.data N = M.shape[0] k = self.k r = self.r # indices of diagonal elements W_diag_idx = numx.arange(N) Q_diag_idx = numx.arange(k) if k > N: err = ('k=%i must be less than or ' 'equal to number of training points N=%i' % (k, N)) raise TrainingException(err) # determines number of output dimensions: if desired_variance # is specified, we need to learn it from the data. Otherwise, # it's easy learn_outdim = False if self.output_dim is None: if self.desired_variance is None: self.output_dim = self.input_dim else: learn_outdim = True # do we need to automatically determine the regularization term? auto_reg = r is None # determine number of output dims, precalculate useful stuff if learn_outdim: Qs, sig2s, nbrss = self._adjust_output_dim() # build the weight matrix #XXX future work: #XXX for faster implementation, W should be a sparse matrix W = numx.zeros((N, N), dtype=self.dtype) if self.verbose: print ' - constructing [%i x %i] weight matrix...' % W.shape for row in range(N): if learn_outdim: Q = Qs[row, :, :] nbrs = nbrss[row, :] else: # ----------------------------------------------- # find k nearest neighbors # ----------------------------------------------- M_Mi = M-M[row] nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1] M_Mi = M_Mi[nbrs] # compute covariance matrix of distances Q = mult(M_Mi, M_Mi.T) # ----------------------------------------------- # compute weight vector based on neighbors # ----------------------------------------------- #Covariance matrix may be nearly singular: # add a diagonal correction to prevent numerical errors if auto_reg: # automatic mode: correction is equal to the sum of # the (d_in-d_out) unused variances (as in deRidder & # Duin) if learn_outdim: sig2 = sig2s[row, :] else: sig2 = svd(M_Mi, compute_uv=0)**2 r = numx.sum(sig2[self.output_dim:]) Q[Q_diag_idx, Q_diag_idx] += r else: # Roweis et al instead use "a correction that # is small compared to the trace" e.g.: # r = 0.001 * float(Q.trace()) # this is equivalent to assuming 0.1% of the variance is unused Q[Q_diag_idx, Q_diag_idx] += r*Q.trace() #solve for weight # weight is w such that sum(Q_ij * w_j) = 1 for all i # XXX refcast is due to numpy bug: floats become double w = self._refcast(numx_linalg.solve(Q, numx.ones(k))) w /= w.sum() #update row of the weight matrix W[nbrs, row] = w if self.verbose: msg = (' - finding [%i x %i] null space of weight matrix\n' ' (may take a while)...' % (self.output_dim, N)) print msg self.W = W.copy() #to find the null space, we need the bottom d+1 # eigenvectors of (W-I).T*(W-I) #Compute this using the svd of (W-I): W[W_diag_idx, W_diag_idx] -= 1. #XXX future work: #XXX use of upcoming ARPACK interface for bottom few eigenvectors #XXX of a sparse matrix will significantly increase the speed #XXX of the next step if self.svd: sig, U = nongeneral_svd(W.T, range=(2, self.output_dim+1)) else: # the following code does the same computation, but uses # symeig, which computes only the required eigenvectors, and # is much faster. However, it could also be more unstable... WW = mult(W, W.T) # regularizes the eigenvalues, does not change the eigenvectors: WW[W_diag_idx, W_diag_idx] += 0.1 sig, U = symeig(WW, range=(2, self.output_dim+1), overwrite=True) self.training_projection = U def _adjust_output_dim(self): # this function is called if we need to compute the number of # output dimensions automatically; some quantities that are # useful later are pre-calculated to spare precious time if self.verbose: print ' - adjusting output dim:' #otherwise, we need to compute output_dim # from desired_variance M = self.data k = self.k N, d_in = M.shape m_est_array = [] Qs = numx.zeros((N, k, k)) sig2s = numx.zeros((N, d_in)) nbrss = numx.zeros((N, k), dtype='i') for row in range(N): #----------------------------------------------- # find k nearest neighbors #----------------------------------------------- M_Mi = M-M[row] nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1] M_Mi = M_Mi[nbrs] # compute covariance matrix of distances Qs[row, :, :] = mult(M_Mi, M_Mi.T) nbrss[row, :] = nbrs #----------------------------------------------- # singular values of M_Mi give the variance: # use this to compute intrinsic dimensionality # at this point #----------------------------------------------- sig2 = (svd(M_Mi, compute_uv=0))**2 sig2s[row, :sig2.shape[0]] = sig2 #----------------------------------------------- # use sig2 to compute intrinsic dimensionality of the # data at this neighborhood. The dimensionality is the # number of eigenvalues needed to sum to the total # desired variance #----------------------------------------------- sig2 /= sig2.sum() S = sig2.cumsum() m_est = S.searchsorted(self.desired_variance) if m_est > 0: m_est += (self.desired_variance-S[m_est-1])/sig2[m_est] else: m_est = self.desired_variance/sig2[m_est] m_est_array.append(m_est) m_est_array = numx.asarray(m_est_array) self.output_dim = int( numx.ceil( numx.median(m_est_array) ) ) if self.verbose: msg = (' output_dim = %i' ' for variance of %.2f' % (self.output_dim, self.desired_variance)) print msg return Qs, sig2s, nbrss def _execute(self, x): #---------------------------------------------------- # similar algorithm to that within self.stop_training() # refer there for notes & comments on code #---------------------------------------------------- N = self.data.shape[0] Nx = x.shape[0] W = numx.zeros((Nx, N), dtype=self.dtype) k, r = self.k, self.r d_out = self.output_dim Q_diag_idx = numx.arange(k) for row in range(Nx): #find nearest neighbors of x in M M_xi = self.data-x[row] nbrs = numx.argsort( (M_xi**2).sum(1) )[:k] M_xi = M_xi[nbrs] #find corrected covariance matrix Q Q = mult(M_xi, M_xi.T) if r is None and k > d_out: sig2 = (svd(M_xi, compute_uv=0))**2 r = numx.sum(sig2[d_out:]) Q[Q_diag_idx, Q_diag_idx] += r if r is not None: Q[Q_diag_idx, Q_diag_idx] += r #solve for weights w = self._refcast(numx_linalg.solve(Q , numx.ones(k))) w /= w.sum() W[row, nbrs] = w #multiply weights by result of SVD from training return numx.dot(W, self.training_projection) @staticmethod def is_trainable(): return True @staticmethod def is_invertible(): return False ######################################################### # Hessian LLE ######################################################### # Modified Gram-Schmidt def _mgs(a): m, n = a.shape v = a.copy() r = numx.zeros((n, n)) for i in range(n): r[i, i] = numx_linalg.norm(v[:, i]) v[:, i] = v[:, i]/r[i, i] for j in range(i+1, n): r[i, j] = mult(v[:, i], v[:, j]) v[:, j] = v[:, j] - r[i, j]*v[:, i] # q is v return v, r class HLLENode(LLENode): """Perform a Hessian Locally Linear Embedding analysis on the data. **Internal variables of interest** ``self.training_projection`` the HLLE projection of the training data (defined when training finishes) ``self.desired_variance`` variance limit used to compute intrinsic dimensionality. Implementation based on algorithm outlined in Donoho, D. L., and Grimes, C., Hessian Eigenmaps: new locally linear embedding techniques for high-dimensional data, Proceedings of the National Academy of Sciences 100(10): 5591-5596, 2003. Original code contributed by: Jake Vanderplas, University of Washington """ #---------------------------------------------------- # Note that many methods ar inherited from LLENode, # including _execute(), _adjust_output_dim(), etc. # The main advantage of the Hessian estimator is to # limit distortions of the input manifold. Once # the model has been trained, it is sufficient (and # much less computationally intensive) to determine # projections for new points using the LLE framework. #---------------------------------------------------- def __init__(self, k, r=0.001, svd=False, verbose=False, input_dim=None, output_dim=None, dtype=None): """ :Keyword arguments: k number of nearest neighbors to use; the node will raise an MDPWarning if k is smaller than k >= 1 + output_dim + output_dim*(output_dim+1)/2, because in this case a less efficient computation must be used, and the ablgorithm can become unstable r regularization constant; as opposed to LLENode, it is not possible to compute this constant automatically; it is only used during execution svd if true, use SVD to compute the projection matrix; SVD is slower but more stable verbose if true, displays information about the progress of the algorithm output_dim number of dimensions to output or a float between 0.0 and 1.0. In the latter case, output_dim specifies the desired fraction of variance to be exaplained, and the final number of output dimensions is known at the end of training (e.g., for 'output_dim=0.95' the algorithm will keep as many dimensions as necessary in order to explain 95% of the input variance) """ LLENode.__init__(self, k, r, svd, verbose, input_dim, output_dim, dtype) def _stop_training(self): Cumulator._stop_training(self) k = self.k M = self.data N = M.shape[0] if k > N: err = ('k=%i must be less than' ' or equal to number of training points N=%i' % (k, N)) raise TrainingException(err) if self.verbose: print 'performing HLLE on %i points in %i dimensions...' % M.shape # determines number of output dimensions: if desired_variance # is specified, we need to learn it from the data. Otherwise, # it's easy learn_outdim = False if self.output_dim is None: if self.desired_variance is None: self.output_dim = self.input_dim else: learn_outdim = True # determine number of output dims, precalculate useful stuff if learn_outdim: Qs, sig2s, nbrss = self._adjust_output_dim() d_out = self.output_dim #dp = d_out + (d_out-1) + (d_out-2) + ... dp = d_out*(d_out+1)/2 if min(k, N) <= d_out: err = ('k=%i and n=%i (number of input data points) must be' ' larger than output_dim=%i' % (k, N, d_out)) raise TrainingException(err) if k < 1+d_out+dp: wrn = ('The number of neighbours, k=%i, is smaller than' ' 1 + output_dim + output_dim*(output_dim+1)/2 = %i,' ' which might result in unstable results.' % (k, 1+d_out+dp)) _warnings.warn(wrn, MDPWarning) #build the weight matrix #XXX for faster implementation, W should be a sparse matrix W = numx.zeros((N, dp*N), dtype=self.dtype) if self.verbose: print ' - constructing [%i x %i] weight matrix...' % W.shape for row in range(N): if learn_outdim: nbrs = nbrss[row, :] else: # ----------------------------------------------- # find k nearest neighbors # ----------------------------------------------- M_Mi = M-M[row] nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1] #----------------------------------------------- # center the neighborhood using the mean #----------------------------------------------- nbrhd = M[nbrs] # this makes a copy nbrhd -= nbrhd.mean(0) #----------------------------------------------- # compute local coordinates # using a singular value decomposition #----------------------------------------------- U, sig, VT = svd(nbrhd) nbrhd = U.T[:d_out] del VT #----------------------------------------------- # build Hessian estimator #----------------------------------------------- Yi = numx.zeros((dp, k), dtype=self.dtype) ct = 0 for i in range(d_out): Yi[ct:ct+d_out-i, :] = nbrhd[i] * nbrhd[i:, :] ct += d_out-i Yi = numx.concatenate([numx.ones((1, k), dtype=self.dtype), nbrhd, Yi], 0) #----------------------------------------------- # orthogonalize linear and quadratic forms # with QR factorization # and make the weights sum to 1 #----------------------------------------------- if k >= 1+d_out+dp: Q, R = numx_linalg.qr(Yi.T) w = Q[:, d_out+1:d_out+1+dp] else: q, r = _mgs(Yi.T) w = q[:, -dp:] S = w.sum(0) #sum along columns #if S[i] is too small, set it equal to 1.0 # this prevents weights from blowing up S[numx.where(numx.absolute(S)<1E-4)] = 1.0 #print w.shape, S.shape, (w/S).shape #print W[nbrs, row*dp:(row+1)*dp].shape W[nbrs, row*dp:(row+1)*dp] = w / S #----------------------------------------------- # To find the null space, we want the # first d+1 eigenvectors of W.T*W # Compute this using an svd of W #----------------------------------------------- if self.verbose: msg = (' - finding [%i x %i] ' 'null space of weight matrix...' % (d_out, N)) print msg #XXX future work: #XXX use of upcoming ARPACK interface for bottom few eigenvectors #XXX of a sparse matrix will significantly increase the speed #XXX of the next step if self.svd: sig, U = nongeneral_svd(W.T, range=(2, d_out+1)) Y = U*numx.sqrt(N) else: WW = mult(W, W.T) # regularizes the eigenvalues, does not change the eigenvectors: W_diag_idx = numx.arange(N) WW[W_diag_idx, W_diag_idx] += 0.01 sig, U = symeig(WW, range=(2, self.output_dim+1), overwrite=True) Y = U*numx.sqrt(N) del WW del W #----------------------------------------------- # Normalize Y # # Alternative way to do it: # we need R = (Y.T*Y)^(-1/2) # do this with an SVD of Y del VT # Y = U*sig*V.T # Y.T*Y = (V*sig.T*U.T) * (U*sig*V.T) # = V * (sig*sig.T) * V.T # = V * sig^2 V.T # so # R = V * sig^-1 * V.T # The code is: # U, sig, VT = svd(Y) # del U # S = numx.diag(sig**-1) # self.training_projection = mult(Y, mult(VT.T, mult(S, VT))) #----------------------------------------------- if self.verbose: print ' - normalizing null space...' C = sqrtm(mult(Y.T, Y)) self.training_projection = mult(Y, C)