Einsum (#1269)

* einsum initial

* fix comma break

* sum axis was wrong

* small cleanups

* python binding

* changed bindings to resemble numpy

* remove todo comment

* comment changes

* add count of operands/inputs

* fail fast if operands list is empty

* ignore comma if no output

* einsum path matching numpy

* getting somewhere with path

* remove print

* it passes the first test

* moved einsum tests to seperate file

* seperated einsum path

* moved einsum naive

* remove space from equation

* fast fail if no operands passed

* update tests and remove printf

* small cleanup

* some more cleanups

* removed python helper file

* ack

* utilize std for finding min in vector

* duplicate def

* remove the tuple as it was unreadable

* moved einsum_naive back to ops

* remaining isn't needed

* avoid creating another set

* cleanup

* greedy path, start of naive einsum

* more einsum

* fix some bugs

* some more fixes, tests pass

* benchmark

* some simplify

* fix einsum and test

Co-authored-by: Angelos Katharopoulos <a_katharopoulos@apple.com>

* add a bunch more tests and fix a bunch more bugs

* some docs nits

---------

Co-authored-by: dc-dc-dc <dgcruz983@gmail.com>
Co-authored-by: Angelos Katharopoulos <a_katharopoulos@apple.com>

benchmarks/python/einsum_bench.py

@@ -0,0 +1,84 @@
+# Copyright © 2024 Apple Inc.
+
+import time
+
+import mlx.core as mx
+import numpy as np
+
+
+def timeit(fn, its=100, args=[]):
+    for _ in range(5):
+        fn(*args)
+    tic = time.perf_counter()
+    for _ in range(its):
+        fn(*args)
+    toc = time.perf_counter()
+    return 1e3 * (toc - tic) / its
+
+
+def time_little_einsum_path():
+    subscripts = "ik,kj->ij"
+    x = mx.ones((32, 32))
+    y = mx.ones((32, 32))
+    mx_time = timeit(mx.einsum_path, args=(subscripts, x, y))
+
+    x = np.array(x)
+    y = np.array(y)
+    np_time = timeit(np.einsum_path, args=(subscripts, x, y))
+    print("Timing little einsum path...")
+    print(f"MLX ... {mx_time:.3f} ms")
+    print(f"NumPy... {np_time:.3f} ms")
+
+
+def time_big_einsum_path():
+    chars = list("abcdefgh")
+    char_to_dim = {c: v for v, c in enumerate(chars)}
+
+    num_inputs = 10
+    inputs = []
+    subscripts = []
+    for _ in range(num_inputs):
+        subscript = np.random.choice(chars, size=5, replace=False).tolist()
+        subscripts.append("".join(subscript))
+        inputs.append(np.ones(list(char_to_dim[c] for c in subscript)))
+    subscripts = ",".join(subscripts)
+
+    np_time = timeit(np.einsum_path, args=(subscripts, *inputs))
+
+    inputs = [mx.array(x) for x in inputs]
+    mx_time = timeit(mx.einsum_path, args=(subscripts, *inputs))
+    print("Timing big einsum path...")
+    print(f"MLX ... {mx_time:.3f} ms")
+    print(f"NumPy... {np_time:.3f} ms")
+
+
+def time_attention():
+    def regular_attention(x):
+        # shape [batch, sequence, num_heads, head_dim]
+        queries, keys, values = x, x, x
+        scores = queries.transpose(0, 2, 1, 3) @ keys.transpose(0, 2, 3, 1)
+        scores = mx.softmax(scores, axis=-1)
+        output = (scores @ values.transpose(0, 2, 1, 3)).swapaxes(1, 2)
+        mx.eval(output)
+
+    def einsum_attention(x):
+        # shape [batch, sequence, num_heads, head_dim]
+        queries, keys, values = x, x, x
+        scores = mx.einsum("itjk,iujk->ijtu", queries, keys)
+        scores = mx.softmax(scores, axis=-1)
+        output = mx.einsum("ijtu,iujk->itjk", scores, values)
+        mx.eval(output)
+
+    x = mx.random.uniform(shape=(8, 512, 32, 128))
+
+    regular_time = timeit(regular_attention, args=(x,))
+    ein_time = timeit(einsum_attention, args=(x,))
+    print("Timing einsum attention...")
+    print(f"Regular ... {regular_time:.3f} ms")
+    print(f"Einsum ... {ein_time:.3f} ms")
+
+
+if __name__ == "__main__":
+    time_little_einsum_path()
+    time_big_einsum_path()
+    time_attention()