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''1a11#   I  ! !  ''1a11#   "    2a5zzS''''[SAZZ[[s=GYGY[[\\\ 2a5zzQN#be**NNOOO IOO  ),RUM)B)B    9  beAh ' ' (  )   "((**        BqE Q EssU2YhH bbxbbb b b    "*,,..       $     X rcddfd S)NrEc4|gfd|DRi|S)Nc&g|] }|Sr r r s rrz.....s!---qqtt---rr )rrtensorskwargsrr s rrz..sJQQN----W---NNN17NNrr rs``rrrs+Tr)rrtrrrrtrreturncpt|gd|DRd|i|\}}t||||\}}|S)a Computes a general dot-product of the input tensors. The following shorthand notation is supported: * When no brackets are found, brackets are placed implicitly around all axes that do not appear in the output. Example: ``a b, b c -> a c`` expands to ``a [b], [b] c -> a c`` * When given two input tensors, the expression of the second input is determined implicitly from the marked axes in the input and output expression. Example: ``a [b] -> a [c]`` expands to ``a b, b c -> a c`` Axes marked multiple times appear only once in the implicit second input expression. Example: ``[a b] -> [a c]`` expands to ``a b, a b c -> a c`` The function additionally passes the ``in_axes``, ``out_axes`` and ``batch_axes`` arguments to tensor factories that can be used to determine the fan-in and fan-out of a neural network layer and initialize weights accordingly (see e.g. `jax.nn.initializers.lecun_normal `_) Args: description: Description string for the operation in einx notation. tensors: Input tensors or tensor factories matching the description string. backend: Backend to use for all operations. If None, determines the backend from the input tensors. Defaults to None. cse: Whether to apply common subexpression elimination to the expressions. Defaults to True. graph: Whether to return the graph representation of the operation instead of computing the result. Defaults to False. **parameters: Additional parameters that specify values for single axes, e.g. ``a=4``. Returns: The result of the dot-product operation if ``graph=False``, otherwise the graph representation of the operation. Examples: Compute an inner product between two vectors: >>> a, b = np.random.uniform(size=(10,)), np.random.uniform(size=(10,)) >>> einx.dot("a, a ->", a, b).shape () Compute a matrix-vector product: >>> a, b = np.random.uniform(size=(10, 10)), np.random.uniform(size=(10,)) >>> einx.dot("a b, b -> a", a, b).shape (10,) >>> einx.dot("a [b] -> a", a, b).shape (10,) >>> einx.dot("a [b->]", a, b).shape (10,) Compute a vector-matrix product: >>> a, b = np.random.uniform(size=(10,)), np.random.uniform(size=(10, 10)) >>> einx.dot("a, a b -> b", a, b).shape (10,) >>> einx.dot("[a] -> [b]", a, b).shape (10,) >>> einx.dot("[a->b]", a, b).shape (10,) Multiply a tensor with a weight matrix: >>> x, w = ( ... np.random.uniform(size=(4, 16, 16, 64)), ... np.random.uniform( ... size=( ... 64, ... 32, ... ) ... ), ... ) >>> einx.dot("b... 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