296 lines
11 KiB
Cython
296 lines
11 KiB
Cython
# Author: Christopher Moody <chrisemoody@gmail.com>
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# Author: Nick Travers <nickt@squareup.com>
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# Implementation by Chris Moody & Nick Travers
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# See http://homepage.tudelft.nl/19j49/t-SNE.html for reference
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# implementations and papers describing the technique
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import numpy as np
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cimport numpy as cnp
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from libc.stdio cimport printf
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from libc.math cimport log
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from libc.stdlib cimport malloc, free
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from libc.time cimport clock, clock_t
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from cython.parallel cimport prange, parallel
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from ..neighbors._quad_tree cimport _QuadTree
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cnp.import_array()
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cdef char* EMPTY_STRING = ""
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# Smallest strictly positive value that can be represented by floating
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# point numbers for different precision levels. This is useful to avoid
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# taking the log of zero when computing the KL divergence.
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cdef float FLOAT32_TINY = np.finfo(np.float32).tiny
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# Useful to void division by zero or divergence to +inf.
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cdef float FLOAT64_EPS = np.finfo(np.float64).eps
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# This is effectively an ifdef statement in Cython
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# It allows us to write printf debugging lines
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# and remove them at compile time
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cdef enum:
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DEBUGFLAG = 0
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cdef float compute_gradient(float[:] val_P,
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float[:, :] pos_reference,
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cnp.int64_t[:] neighbors,
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cnp.int64_t[:] indptr,
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float[:, :] tot_force,
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_QuadTree qt,
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float theta,
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int dof,
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long start,
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bint compute_error,
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int num_threads) noexcept nogil:
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# Having created the tree, calculate the gradient
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# in two components, the positive and negative forces
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cdef:
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long i, coord
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int ax
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long n_samples = pos_reference.shape[0]
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int n_dimensions = qt.n_dimensions
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clock_t t1 = 0, t2 = 0
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double sQ
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float error
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int take_timing = 1 if qt.verbose > 15 else 0
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if qt.verbose > 11:
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printf("[t-SNE] Allocating %li elements in force arrays\n",
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n_samples * n_dimensions * 2)
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cdef float* neg_f = <float*> malloc(sizeof(float) * n_samples * n_dimensions)
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cdef float* pos_f = <float*> malloc(sizeof(float) * n_samples * n_dimensions)
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if take_timing:
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t1 = clock()
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sQ = compute_gradient_negative(pos_reference, neg_f, qt, dof, theta, start,
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num_threads)
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if take_timing:
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t2 = clock()
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printf("[t-SNE] Computing negative gradient: %e ticks\n", ((float) (t2 - t1)))
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if take_timing:
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t1 = clock()
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error = compute_gradient_positive(val_P, pos_reference, neighbors, indptr,
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pos_f, n_dimensions, dof, sQ, start,
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qt.verbose, compute_error, num_threads)
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if take_timing:
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t2 = clock()
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printf("[t-SNE] Computing positive gradient: %e ticks\n",
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((float) (t2 - t1)))
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for i in prange(start, n_samples, nogil=True, num_threads=num_threads,
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schedule='static'):
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for ax in range(n_dimensions):
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coord = i * n_dimensions + ax
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tot_force[i, ax] = pos_f[coord] - (neg_f[coord] / sQ)
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free(neg_f)
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free(pos_f)
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return error
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cdef float compute_gradient_positive(float[:] val_P,
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float[:, :] pos_reference,
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cnp.int64_t[:] neighbors,
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cnp.int64_t[:] indptr,
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float* pos_f,
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int n_dimensions,
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int dof,
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double sum_Q,
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cnp.int64_t start,
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int verbose,
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bint compute_error,
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int num_threads) noexcept nogil:
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# Sum over the following expression for i not equal to j
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# grad_i = p_ij (1 + ||y_i - y_j||^2)^-1 (y_i - y_j)
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# This is equivalent to compute_edge_forces in the authors' code
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# It just goes over the nearest neighbors instead of all the data points
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# (unlike the non-nearest neighbors version of `compute_gradient_positive')
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cdef:
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int ax
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long i, j, k
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long n_samples = indptr.shape[0] - 1
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float C = 0.0
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float dij, qij, pij
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float exponent = (dof + 1.0) / 2.0
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float float_dof = (float) (dof)
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float* buff
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clock_t t1 = 0, t2 = 0
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float dt
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if verbose > 10:
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t1 = clock()
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with nogil, parallel(num_threads=num_threads):
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# Define private buffer variables
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buff = <float *> malloc(sizeof(float) * n_dimensions)
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for i in prange(start, n_samples, schedule='static'):
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# Init the gradient vector
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for ax in range(n_dimensions):
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pos_f[i * n_dimensions + ax] = 0.0
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# Compute the positive interaction for the nearest neighbors
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for k in range(indptr[i], indptr[i+1]):
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j = neighbors[k]
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dij = 0.0
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pij = val_P[k]
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for ax in range(n_dimensions):
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buff[ax] = pos_reference[i, ax] - pos_reference[j, ax]
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dij += buff[ax] * buff[ax]
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qij = float_dof / (float_dof + dij)
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if dof != 1: # i.e. exponent != 1
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qij = qij ** exponent
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dij = pij * qij
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# only compute the error when needed
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if compute_error:
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qij = qij / sum_Q
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C += pij * log(max(pij, FLOAT32_TINY) / max(qij, FLOAT32_TINY))
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for ax in range(n_dimensions):
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pos_f[i * n_dimensions + ax] += dij * buff[ax]
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free(buff)
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if verbose > 10:
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t2 = clock()
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dt = ((float) (t2 - t1))
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printf("[t-SNE] Computed error=%1.4f in %1.1e ticks\n", C, dt)
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return C
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cdef double compute_gradient_negative(float[:, :] pos_reference,
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float* neg_f,
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_QuadTree qt,
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int dof,
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float theta,
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long start,
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int num_threads) noexcept nogil:
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cdef:
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int ax
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int n_dimensions = qt.n_dimensions
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int offset = n_dimensions + 2
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long i, j, idx
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long n_samples = pos_reference.shape[0]
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long n = n_samples - start
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long dta = 0
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long dtb = 0
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float size, dist2s, mult
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float exponent = (dof + 1.0) / 2.0
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float float_dof = (float) (dof)
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double qijZ, sum_Q = 0.0
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float* force
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float* neg_force
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float* pos
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clock_t t1 = 0, t2 = 0, t3 = 0
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int take_timing = 1 if qt.verbose > 20 else 0
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with nogil, parallel(num_threads=num_threads):
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# Define thread-local buffers
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summary = <float*> malloc(sizeof(float) * n * offset)
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pos = <float *> malloc(sizeof(float) * n_dimensions)
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force = <float *> malloc(sizeof(float) * n_dimensions)
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neg_force = <float *> malloc(sizeof(float) * n_dimensions)
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for i in prange(start, n_samples, schedule='static'):
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# Clear the arrays
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for ax in range(n_dimensions):
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force[ax] = 0.0
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neg_force[ax] = 0.0
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pos[ax] = pos_reference[i, ax]
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# Find which nodes are summarizing and collect their centers of mass
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# deltas, and sizes, into vectorized arrays
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if take_timing:
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t1 = clock()
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idx = qt.summarize(pos, summary, theta*theta)
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if take_timing:
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t2 = clock()
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# Compute the t-SNE negative force
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# for the digits dataset, walking the tree
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# is about 10-15x more expensive than the
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# following for loop
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for j in range(idx // offset):
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dist2s = summary[j * offset + n_dimensions]
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size = summary[j * offset + n_dimensions + 1]
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qijZ = float_dof / (float_dof + dist2s) # 1/(1+dist)
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if dof != 1: # i.e. exponent != 1
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qijZ = qijZ ** exponent
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sum_Q += size * qijZ # size of the node * q
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mult = size * qijZ * qijZ
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for ax in range(n_dimensions):
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neg_force[ax] += mult * summary[j * offset + ax]
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if take_timing:
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t3 = clock()
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for ax in range(n_dimensions):
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neg_f[i * n_dimensions + ax] = neg_force[ax]
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if take_timing:
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dta += t2 - t1
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dtb += t3 - t2
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free(pos)
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free(force)
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free(neg_force)
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free(summary)
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if take_timing:
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printf("[t-SNE] Tree: %li clock ticks | ", dta)
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printf("Force computation: %li clock ticks\n", dtb)
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# Put sum_Q to machine EPSILON to avoid divisions by 0
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sum_Q = max(sum_Q, FLOAT64_EPS)
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return sum_Q
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def gradient(float[:] val_P,
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float[:, :] pos_output,
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cnp.int64_t[:] neighbors,
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cnp.int64_t[:] indptr,
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float[:, :] forces,
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float theta,
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int n_dimensions,
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int verbose,
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int dof=1,
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long skip_num_points=0,
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bint compute_error=1,
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int num_threads=1):
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# This function is designed to be called from external Python
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# it passes the 'forces' array by reference and fills that's array
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# up in-place
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cdef float C
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cdef int n
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n = pos_output.shape[0]
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assert val_P.itemsize == 4
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assert pos_output.itemsize == 4
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assert forces.itemsize == 4
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m = "Forces array and pos_output shapes are incompatible"
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assert n == forces.shape[0], m
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m = "Pij and pos_output shapes are incompatible"
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assert n == indptr.shape[0] - 1, m
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if verbose > 10:
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printf("[t-SNE] Initializing tree of n_dimensions %i\n", n_dimensions)
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cdef _QuadTree qt = _QuadTree(pos_output.shape[1], verbose)
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if verbose > 10:
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printf("[t-SNE] Inserting %li points\n", pos_output.shape[0])
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qt.build_tree(pos_output)
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if verbose > 10:
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# XXX: format hack to workaround lack of `const char *` type
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# in the generated C code that triggers error with gcc 4.9
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# and -Werror=format-security
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printf("[t-SNE] Computing gradient\n%s", EMPTY_STRING)
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C = compute_gradient(val_P, pos_output, neighbors, indptr, forces,
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qt, theta, dof, skip_num_points, compute_error,
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num_threads)
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if verbose > 10:
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# XXX: format hack to workaround lack of `const char *` type
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# in the generated C code
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# and -Werror=format-security
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printf("[t-SNE] Checking tree consistency\n%s", EMPTY_STRING)
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m = "Tree consistency failed: unexpected number of points on the tree"
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assert qt.cells[0].cumulative_size == qt.n_points, m
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if not compute_error:
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C = np.nan
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return C
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