# Copyright 2016 The TensorFlow Authors. All Rights Reserved. # # Licensed under the Apache License, Version 2.0 (the "License"); # you may not use this file except in compliance with the License. # You may obtain a copy of the License at # # http://www.apache.org/licenses/LICENSE-2.0 # # Unless required by applicable law or agreed to in writing, software # distributed under the License is distributed on an "AS IS" BASIS, # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. # See the License for the specific language governing permissions and # limitations under the License. # ============================================================================== # Functions "ndtr" and "ndtri" are derived from calculations made in: # https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html # In the following email exchange, the author gives his consent to redistribute # derived works under an Apache 2.0 license. # # From: Stephen Moshier # Date: Sat, Jun 9, 2018 at 2:36 PM # Subject: Re: Licensing cephes under Apache (BSD-like) license. # To: rif # # # # Hello Rif, # # Yes, Google may distribute Cephes files under the Apache 2 license. # # If clarification is needed, I do not favor BSD over other free licenses. # I would agree that Apache 2 seems to cover the concern you mentioned # about sublicensees. # # Best wishes for good luck with your projects! # Steve Moshier # # # # On Thu, 31 May 2018, rif wrote: # # > Hello Steve. # > My name is Rif. I work on machine learning software at Google. # > # > Your cephes software continues to be incredibly useful and widely used. I # > was wondering whether it would be permissible for us to use the Cephes code # > under the Apache 2.0 license, which is extremely similar in permissions to # > the BSD license (Wikipedia comparisons). This would be quite helpful to us # > in terms of avoiding multiple licenses on software. # > # > I'm sorry to bother you with this (I can imagine you're sick of hearing # > about this by now), but I want to be absolutely clear we're on the level and # > not misusing your important software. In former conversation with Eugene # > Brevdo (ebrevdo@google.com), you wrote "If your licensing is similar to BSD, # > the formal way that has been handled is simply to add a statement to the # > effect that you are incorporating the Cephes software by permission of the # > author." I wanted to confirm that (a) we could use the Apache license, (b) # > that we don't need to (and probably you don't want to) keep getting # > contacted about individual uses, because your intent is generally to allow # > this software to be reused under "BSD-like" license, and (c) you're OK # > letting incorporators decide whether a license is sufficiently BSD-like? # > # > Best, # > # > rif # > # > # > """Special Math Ops.""" import numpy as np from tensorflow.python.framework import constant_op from tensorflow.python.framework import ops from tensorflow.python.ops import array_ops from tensorflow.python.ops import math_ops __all__ = [ "erfinv", "ndtr", "ndtri", "log_ndtr", "log_cdf_laplace", ] # log_ndtr uses different functions over the ranges # (-infty, lower](lower, upper](upper, infty) # Lower bound values were chosen by examining where the support of ndtr # appears to be zero, relative to scipy's (which is always 64bit). They were # then made more conservative just to be safe. (Conservative means use the # expansion more than we probably need to.) See `NdtrTest` in # special_math_test.py. LOGNDTR_FLOAT64_LOWER = np.array(-20, np.float64) LOGNDTR_FLOAT32_LOWER = np.array(-10, np.float32) # Upper bound values were chosen by examining for which values of 'x' # Log[cdf(x)] is 0, after which point we need to use the approximation # Log[cdf(x)] = Log[1 - cdf(-x)] approx -cdf(-x). We chose a value slightly # conservative, meaning we use the approximation earlier than needed. LOGNDTR_FLOAT64_UPPER = np.array(8, np.float64) LOGNDTR_FLOAT32_UPPER = np.array(5, np.float32) def ndtr(x, name="ndtr"): """Normal distribution function. Returns the area under the Gaussian probability density function, integrated from minus infinity to x: ``` 1 / x ndtr(x) = ---------- | exp(-0.5 t**2) dt sqrt(2 pi) /-inf = 0.5 (1 + erf(x / sqrt(2))) = 0.5 erfc(x / sqrt(2)) ``` Args: x: `Tensor` of type `float32`, `float64`. name: Python string. A name for the operation (default="ndtr"). Returns: ndtr: `Tensor` with `dtype=x.dtype`. Raises: TypeError: if `x` is not floating-type. """ with ops.name_scope(name, values=[x]): x = ops.convert_to_tensor(x, name="x") if x.dtype.as_numpy_dtype not in [np.float32, np.float64]: raise TypeError( "x.dtype=%s is not handled, see docstring for supported types." % x.dtype) return _ndtr(x) def _ndtr(x): """Implements ndtr core logic.""" half_sqrt_2 = constant_op.constant( 0.5 * np.sqrt(2.), dtype=x.dtype, name="half_sqrt_2") w = x * half_sqrt_2 z = math_ops.abs(w) y = array_ops.where_v2( math_ops.less(z, half_sqrt_2), 1. + math_ops.erf(w), array_ops.where_v2( math_ops.greater(w, 0.), 2. - math_ops.erfc(z), math_ops.erfc(z))) return 0.5 * y def ndtri(p, name="ndtri"): """The inverse of the CDF of the Normal distribution function. Returns x such that the area under the pdf from minus infinity to x is equal to p. A piece-wise rational approximation is done for the function. This is a port of the implementation in netlib. Args: p: `Tensor` of type `float32`, `float64`. name: Python string. A name for the operation (default="ndtri"). Returns: x: `Tensor` with `dtype=p.dtype`. Raises: TypeError: if `p` is not floating-type. """ with ops.name_scope(name, values=[p]): p = ops.convert_to_tensor(p, name="p") if p.dtype.as_numpy_dtype not in [np.float32, np.float64]: raise TypeError( "p.dtype=%s is not handled, see docstring for supported types." % p.dtype) return _ndtri(p) def _ndtri(p): """Implements ndtri core logic.""" # Constants used in piece-wise rational approximations. Taken from the cephes # library: # https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html p0 = [ -1.23916583867381258016E0, 1.39312609387279679503E1, -5.66762857469070293439E1, 9.80010754185999661536E1, -5.99633501014107895267E1 ] q0 = [ -1.18331621121330003142E0, 1.59056225126211695515E1, -8.20372256168333339912E1, 2.00260212380060660359E2, -2.25462687854119370527E2, 8.63602421390890590575E1, 4.67627912898881538453E0, 1.95448858338141759834E0, 1.0 ] p1 = [ -8.57456785154685413611E-4, -3.50424626827848203418E-2, -1.40256079171354495875E-1, 2.18663306850790267539E0, 1.46849561928858024014E1, 4.40805073893200834700E1, 5.71628192246421288162E1, 3.15251094599893866154E1, 4.05544892305962419923E0 ] q1 = [ -9.33259480895457427372E-4, -3.80806407691578277194E-2, -1.42182922854787788574E-1, 2.50464946208309415979E0, 1.50425385692907503408E1, 4.13172038254672030440E1, 4.53907635128879210584E1, 1.57799883256466749731E1, 1.0 ] p2 = [ 6.23974539184983293730E-9, 2.65806974686737550832E-6, 3.01581553508235416007E-4, 1.23716634817820021358E-2, 2.01485389549179081538E-1, 1.33303460815807542389E0, 3.93881025292474443415E0, 6.91522889068984211695E0, 3.23774891776946035970E0 ] q2 = [ 6.79019408009981274425E-9, 2.89247864745380683936E-6, 3.28014464682127739104E-4, 1.34204006088543189037E-2, 2.16236993594496635890E-1, 1.37702099489081330271E0, 3.67983563856160859403E0, 6.02427039364742014255E0, 1.0 ] def _create_polynomial(var, coeffs): """Compute n_th order polynomial via Horner's method.""" coeffs = np.array(coeffs, var.dtype.as_numpy_dtype) if not coeffs.size: return array_ops.zeros_like(var) return coeffs[0] + _create_polynomial(var, coeffs[1:]) * var maybe_complement_p = array_ops.where_v2(p > -np.expm1(-2.), 1. - p, p) # Write in an arbitrary value in place of 0 for p since 0 will cause NaNs # later on. The result from the computation when p == 0 is not used so any # number that doesn't result in NaNs is fine. sanitized_mcp = array_ops.where_v2( maybe_complement_p <= 0., array_ops.fill(array_ops.shape(p), np.array(0.5, p.dtype.as_numpy_dtype)), maybe_complement_p) # Compute x for p > exp(-2): x/sqrt(2pi) = w + w**3 P0(w**2)/Q0(w**2). w = sanitized_mcp - 0.5 ww = w ** 2 x_for_big_p = w + w * ww * (_create_polynomial(ww, p0) / _create_polynomial(ww, q0)) x_for_big_p *= -np.sqrt(2. * np.pi) # Compute x for p <= exp(-2): x = z - log(z)/z - (1/z) P(1/z) / Q(1/z), # where z = sqrt(-2. * log(p)), and P/Q are chosen between two different # arrays based on whether p < exp(-32). z = math_ops.sqrt(-2. * math_ops.log(sanitized_mcp)) first_term = z - math_ops.log(z) / z second_term_small_p = ( _create_polynomial(1. / z, p2) / _create_polynomial(1. / z, q2) / z) second_term_otherwise = ( _create_polynomial(1. / z, p1) / _create_polynomial(1. / z, q1) / z) x_for_small_p = first_term - second_term_small_p x_otherwise = first_term - second_term_otherwise x = array_ops.where_v2( sanitized_mcp > np.exp(-2.), x_for_big_p, array_ops.where_v2(z >= 8.0, x_for_small_p, x_otherwise)) x = array_ops.where_v2(p > 1. - np.exp(-2.), x, -x) infinity_scalar = constant_op.constant(np.inf, dtype=p.dtype) infinity = array_ops.fill(array_ops.shape(p), infinity_scalar) x_nan_replaced = array_ops.where_v2(p <= 0.0, -infinity, array_ops.where_v2(p >= 1.0, infinity, x)) return x_nan_replaced def log_ndtr(x, series_order=3, name="log_ndtr"): """Log Normal distribution function. For details of the Normal distribution function see `ndtr`. This function calculates `(log o ndtr)(x)` by either calling `log(ndtr(x))` or using an asymptotic series. Specifically: - For `x > upper_segment`, use the approximation `-ndtr(-x)` based on `log(1-x) ~= -x, x << 1`. - For `lower_segment < x <= upper_segment`, use the existing `ndtr` technique and take a log. - For `x <= lower_segment`, we use the series approximation of erf to compute the log CDF directly. The `lower_segment` is set based on the precision of the input: ``` lower_segment = { -20, x.dtype=float64 { -10, x.dtype=float32 upper_segment = { 8, x.dtype=float64 { 5, x.dtype=float32 ``` When `x < lower_segment`, the `ndtr` asymptotic series approximation is: ``` ndtr(x) = scale * (1 + sum) + R_N scale = exp(-0.5 x**2) / (-x sqrt(2 pi)) sum = Sum{(-1)^n (2n-1)!! / (x**2)^n, n=1:N} R_N = O(exp(-0.5 x**2) (2N+1)!! / |x|^{2N+3}) ``` where `(2n-1)!! = (2n-1) (2n-3) (2n-5) ... (3) (1)` is a [double-factorial](https://en.wikipedia.org/wiki/Double_factorial). Args: x: `Tensor` of type `float32`, `float64`. series_order: Positive Python `integer`. Maximum depth to evaluate the asymptotic expansion. This is the `N` above. name: Python string. A name for the operation (default="log_ndtr"). Returns: log_ndtr: `Tensor` with `dtype=x.dtype`. Raises: TypeError: if `x.dtype` is not handled. TypeError: if `series_order` is a not Python `integer.` ValueError: if `series_order` is not in `[0, 30]`. """ if not isinstance(series_order, int): raise TypeError("series_order must be a Python integer.") if series_order < 0: raise ValueError("series_order must be non-negative.") if series_order > 30: raise ValueError("series_order must be <= 30.") with ops.name_scope(name, values=[x]): x = ops.convert_to_tensor(x, name="x") if x.dtype.as_numpy_dtype == np.float64: lower_segment = LOGNDTR_FLOAT64_LOWER upper_segment = LOGNDTR_FLOAT64_UPPER elif x.dtype.as_numpy_dtype == np.float32: lower_segment = LOGNDTR_FLOAT32_LOWER upper_segment = LOGNDTR_FLOAT32_UPPER else: raise TypeError("x.dtype=%s is not supported." % x.dtype) # The basic idea here was ported from: # https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html # We copy the main idea, with a few changes # * For x >> 1, and X ~ Normal(0, 1), # Log[P[X < x]] = Log[1 - P[X < -x]] approx -P[X < -x], # which extends the range of validity of this function. # * We use one fixed series_order for all of 'x', rather than adaptive. # * Our docstring properly reflects that this is an asymptotic series, not a # Taylor series. We also provided a correct bound on the remainder. # * We need to use the max/min in the _log_ndtr_lower arg to avoid nan when # x=0. This happens even though the branch is unchosen because when x=0 # the gradient of a select involves the calculation 1*dy+0*(-inf)=nan # regardless of whether dy is finite. Note that the minimum is a NOP if # the branch is chosen. return array_ops.where_v2( math_ops.greater(x, upper_segment), -_ndtr(-x), # log(1-x) ~= -x, x << 1 # pylint: disable=invalid-unary-operand-type array_ops.where_v2( math_ops.greater(x, lower_segment), math_ops.log(_ndtr(math_ops.maximum(x, lower_segment))), _log_ndtr_lower(math_ops.minimum(x, lower_segment), series_order))) def _log_ndtr_lower(x, series_order): """Asymptotic expansion version of `Log[cdf(x)]`, appropriate for `x<<-1`.""" x_2 = math_ops.square(x) # Log of the term multiplying (1 + sum) log_scale = -0.5 * x_2 - math_ops.log(-x) - 0.5 * np.log(2. * np.pi) return log_scale + math_ops.log(_log_ndtr_asymptotic_series(x, series_order)) def _log_ndtr_asymptotic_series(x, series_order): """Calculates the asymptotic series used in log_ndtr.""" dtype = x.dtype.as_numpy_dtype if series_order <= 0: return np.array(1, dtype) x_2 = math_ops.square(x) even_sum = array_ops.zeros_like(x) odd_sum = array_ops.zeros_like(x) x_2n = x_2 # Start with x^{2*1} = x^{2*n} with n = 1. for n in range(1, series_order + 1): y = np.array(_double_factorial(2 * n - 1), dtype) / x_2n if n % 2: odd_sum += y else: even_sum += y x_2n *= x_2 return 1. + even_sum - odd_sum def erfinv(x, name="erfinv"): """The inverse function for erf, the error function. Args: x: `Tensor` of type `float32`, `float64`. name: Python string. A name for the operation (default="erfinv"). Returns: x: `Tensor` with `dtype=x.dtype`. Raises: TypeError: if `x` is not floating-type. """ with ops.name_scope(name, values=[x]): x = ops.convert_to_tensor(x, name="x") if x.dtype.as_numpy_dtype not in [np.float32, np.float64]: raise TypeError( "x.dtype=%s is not handled, see docstring for supported types." % x.dtype) return ndtri((x + 1.0) / 2.0) / np.sqrt(2) def _double_factorial(n): """The double factorial function for small Python integer `n`.""" return np.prod(np.arange(n, 1, -2)) def log_cdf_laplace(x, name="log_cdf_laplace"): """Log Laplace distribution function. This function calculates `Log[L(x)]`, where `L(x)` is the cumulative distribution function of the Laplace distribution, i.e. ```L(x) := 0.5 * int_{-infty}^x e^{-|t|} dt``` For numerical accuracy, `L(x)` is computed in different ways depending on `x`, ``` x <= 0: Log[L(x)] = Log[0.5] + x, which is exact 0 < x: Log[L(x)] = Log[1 - 0.5 * e^{-x}], which is exact ``` Args: x: `Tensor` of type `float32`, `float64`. name: Python string. A name for the operation (default="log_ndtr"). Returns: `Tensor` with `dtype=x.dtype`. Raises: TypeError: if `x.dtype` is not handled. """ with ops.name_scope(name, values=[x]): x = ops.convert_to_tensor(x, name="x") # For x < 0, L(x) = 0.5 * exp{x} exactly, so Log[L(x)] = log(0.5) + x. lower_solution = -np.log(2.) + x # safe_exp_neg_x = exp{-x} for x > 0, but is # bounded above by 1, which avoids # log[1 - 1] = -inf for x = log(1/2), AND # exp{-x} --> inf, for x << -1 safe_exp_neg_x = math_ops.exp(-math_ops.abs(x)) # log1p(z) = log(1 + z) approx z for |z| << 1. This approximation is used # internally by log1p, rather than being done explicitly here. upper_solution = math_ops.log1p(-0.5 * safe_exp_neg_x) return array_ops.where_v2(x < 0., lower_solution, upper_solution)