def PerformPositionOperations(pos, positions=None):
"""Gets pos and returns (q1, ..., q5)."""
succ_keys = positions[:-1, :]
succ_values = positions[1:, :]
subtract_1_keys = positions[1:, :]
subtract_1_values = positions[:-1, :]
l = int(positions.shape[0]) // 2
add_keys = np.array([
np.concatenate([positions[i, :], positions[j, :]]) for i in range(l)
for j in range(l)
])
add_values = np.array(
[positions[i + j, :] for i in range(l) for j in range(l)])
# TODO(lukaszkaiser): try this below: "for j in range(i) for i in range(2*l)"
sub_keys = np.array([
np.concatenate([positions[i, :], positions[j, :]]) for j in range(l)
for i in range(l)
])
sub_values = np.array(
[positions[max(i - j, 0), :] for j in range(l) for i in range(l)])
query_types = [
QueryPositionKV(),
QueryPositionKV(keys=succ_keys, values=succ_values),
QueryPositionKV(keys=subtract_1_keys, values=subtract_1_values),
QueryPositionKV(keys=add_keys, values=add_values, binary=True),
QueryPositionKV(keys=sub_keys, values=sub_values, binary=True)
]
return [qt @ pos for qt in query_types] # pylint: disable=syntax-error
def forward_with_state(self, x, weights, state, rng):
batch_size, length = x.shape[0], x.shape[1]
max_pos = min(self._bases)**self._n_digits
rng1, rng2, rng3 = math.random.split(rng, 3)
assert length < max_pos, 'length (%d) >= max_pos (%d)' % (length,
max_pos)
positions = jnp.arange(0, length)[None, :]
if self._mode == 'train':
# In 1% of training cases still start from 0 to be exactly as in eval.
start_from_nonzero = jax.random.randint(
rng1, (batch_size, ), 0, self._start_from_zero_one_in)
start_from_nonzero = jnp.minimum(1, start_from_nonzero)
random_start = jax.random.randint(rng2, (batch_size, ), 0,
max_pos - length)
random_start *= start_from_nonzero
positions += random_start[:, None]
res = []
for bn, base in enumerate(self._bases):
pos_embeddings = []
cur_positions = positions
for i in range(self._n_digits):
cur_indices = jnp.mod(cur_positions, base)
cur_positions = cur_positions // base
s = weights[bn][i]
pos_embeddings.append(
cur_indices.astype(jnp.float32)[:, :, None] * s)
embeddings = jnp.concatenate(pos_embeddings, axis=-1)
if self._mode == 'train':
base_dropout = jax.random.randint(rng3, (batch_size, ), 0,
self._base_dropout_one_in)
base_dropout = jnp.minimum(1, base_dropout).astype(jnp.float32)
embeddings *= base_dropout[:, None, None]
res.append(embeddings)
res = sum(res) + jnp.zeros_like(x)
return jnp.concatenate([x, res], axis=-1), state
def Interleave(inputs, **unused_kwargs):
"""Interleaves and flattens two serialized sequences.
The first sequence can be longer by 1 than the second one. This is so we can
interleave sequences of observations and actions, when there's 1 extra
observation at the end.
For serialized sequences [[x_1_1, ..., x_1_R1], ..., [x_L1_1, ..., x_L1_R1]]
and [[y_1_1, ..., y_1_R2], ..., [y_L2_1, ..., y_L2_R2]], where L1 = L2 + 1,
the result is [x_1_1, ..., x_1_R1, y_1_1, ..., y_1_R2, ..., x_L2_1, ...,
x_L2_R1, y_L2_1, ..., y_L2_R2, x_L1_1, ..., x_L1_R1] (batch dimension omitted
for clarity).
Args:
inputs: Pair of sequences of shapes (B, L1, R1) and (B, L2, R2), where B
is batch size, L* is the length of the sequence and R* is the
representation length of each element in the sequence.
Returns:
Interleaved sequence of shape (B, L1 * R1 + L2 * R2).
"""
(x, y) = inputs
(batch_size, _, _) = x.shape
(_, length, _) = y.shape
assert x.shape[1] in (length, length + 1)
reprs = np.concatenate((x[:, :length], y), axis=2)
reprs = np.reshape(reprs, (batch_size, -1))
remainder = np.reshape(x[:, length:], (batch_size, -1))
return np.concatenate((reprs, remainder), axis=1)
def interleave(x, y):
(batch_size, _, _) = x.shape
(_, length, _) = y.shape
assert x.shape[1] in (length, length + 1)
reprs = jnp.concatenate((x[:, :length], y), axis=2)
reprs = jnp.reshape(reprs, (batch_size, -1))
remainder = jnp.reshape(x[:, length:], (batch_size, -1))
return jnp.concatenate((reprs, remainder), axis=1)
def reverse(self, output, weights=(), state=(), new_state=(), **kwargs):
del weights, kwargs
x1_split = []
x2_split = []
for y in output:
y1, y2 = np.split(y, 2, -1)
x1_split.append(y1)
x2_split.append(y2)
x1 = np.concatenate(x1_split, self._axis)
x2 = np.concatenate(x2_split, self._axis)
return (x1, x2)
def forward(self, inputs, weights):
x, gru_state = inputs
# Dense layer on the concatenation of x and h.
w1, b1, w2, b2 = weights
y = jnp.dot(jnp.concatenate([x, gru_state], axis=-1), w1) + b1
# Update and reset gates.
u, r = jnp.split(math.sigmoid(y), 2, axis=-1)
# Candidate.
c = jnp.dot(jnp.concatenate([x, r * gru_state], axis=-1), w2) + b2
new_gru_state = u * gru_state + (1 - u) * jnp.tanh(c)
return new_gru_state, new_gru_state
def forward_with_state(self,
inputs,
weights=layer_base.EMPTY_WEIGHTS,
state=layer_base.EMPTY_STATE,
rng=None,
**kwargs):
depth = inputs.shape[-1]
if self._mode == 'predict':
emb = self._get_embeddings(t=state)
emb = emb[:, np.newaxis, :]
state = state + 1
else:
input_len = inputs.shape[-2]
emb = self._get_embeddings(t=np.arange(input_len, dtype=np.int32))
# Leave batch axis as 1 for broadcasting:
emb = emb[np.newaxis, :, :]
emb = np.broadcast_to(emb, inputs.shape[:-1] + (3, ))
# Replace the last num_features channels of input.
inputs = np.concatenate([inputs[..., :-self.num_features], emb], -1)
if inputs.shape[-1] > depth:
logging.warning('dropping feature(s): %d down to %d',
inputs.shape[-1], depth)
inputs = inputs[..., -depth:]
assert inputs.shape[-1] == depth, inputs.shape
return inputs, state
def test_weights_state(self):
layer = base.Fn(
'2in2out',
lambda x, y: (x + y, jnp.concatenate([x, y], axis=0)), n_out=2)
weights, state = layer.new_weights_and_state(None)
self.assertEmpty(weights)
self.assertEmpty(state)
def forward_with_state(self,
inputs,
weights=base.EMPTY_WEIGHTS,
state=base.EMPTY_STATE,
rng=None,
**kwargs):
embs = []
for ax_emb in weights:
ax_emb = np.broadcast_to(ax_emb, (inputs.shape[0], ) +
self._shape + (ax_emb.shape[-1], ))
embs.append(ax_emb)
emb = np.concatenate(embs, -1)
if self._mode == 'predict':
assert self._dropout == 0.0
emb = np.reshape(emb, (inputs.shape[0], -1, emb.shape[-1]))
return inputs + emb[:, state, :][:, None, :], state + 1
elif self._dropout == 0:
return inputs + np.reshape(emb, inputs.shape), state
else:
noise_shape = list(emb.shape)
for dim in self._dropout_broadcast_dims:
noise_shape[dim] = 1
keep_prob = 1.0 - self._dropout
if math.backend_name() == 'jax':
keep_prob = jax.lax.tie_in(
inputs, np.full((), keep_prob, dtype=inputs.dtype))
keep = math.random.bernoulli(rng, keep_prob, tuple(noise_shape))
multiplier = keep.astype(inputs.dtype) / keep_prob
return inputs + np.reshape(emb * multiplier, inputs.shape), state
def CombineHeadsPos(x, n_heads=1, **unused_kwargs):
"""Mix x = (x0, p0, ..., xH, pH) into (x0, ...., xH), p_combined.
The positions are averaged as vectors.
Args:
x: input vector, concatenated (x0, p0, ..., xH, pH).
n_heads: number of heads.
Returns:
the vector with combined xs and one with combined positions.
"""
seqlen = x.shape[1]
d_head = x.shape[2]
x = np.reshape(x, (-1, n_heads, seqlen, d_head))
x = np.transpose(x, (0, 2, 1, 3)) # -> n_batch, seqlen, n_heads, d_head
x = np.reshape(x, (-1, seqlen, n_heads * d_head))
head_size = int(d_head) - POS_VECTOR_SIZE
res, positions, idx = [], [], 0
for _ in range(n_heads):
res.append(x[:, :, idx:idx + head_size])
idx += head_size
positions.append(x[:, :, idx:idx + POS_VECTOR_SIZE])
idx += POS_VECTOR_SIZE
combined_position = sum(positions) / float(len(positions))
return np.concatenate(res, axis=-1), combined_position
def forward(self, inputs, weights):
state = self.state
depth = inputs.shape[-1]
if self._mode == 'predict':
emb = self._get_embeddings(t=state)
emb = emb[:, jnp.newaxis, :]
state = state + 1
else:
input_len = inputs.shape[-2]
emb = self._get_embeddings(
t=jnp.arange(input_len, dtype=jnp.int32))
# Leave batch axis as 1 for broadcasting:
emb = emb[jnp.newaxis, :, :]
emb = jnp.broadcast_to(emb, inputs.shape[:-1] + (3, ))
# Replace the last num_features channels of input.
inputs = jnp.concatenate([inputs[..., :-self.num_features], emb], -1)
if inputs.shape[-1] > depth:
logging.warning('dropping feature(s): %d down to %d',
inputs.shape[-1], depth)
inputs = inputs[..., -depth:]
assert inputs.shape[-1] == depth, inputs.shape
self.state = state
return inputs
def reverse(self, output, weights=(), state=(), new_state=(), **kwargs):
del weights, kwargs
if not isinstance(output, (list, tuple)):
output = [output]
x1_split = []
x2_split = []
for y in output:
y1, y2 = np.split(y, 2, -1)
x1_split.append(y1)
x2_split.append(y2)
x1 = np.concatenate(x1_split, self._axis)
x2 = np.concatenate(x2_split, self._axis)
return (x1, x2)
def forward(self, inputs, weights):
x, lstm_state = inputs
# LSTM state consists of c and h.
c, h = jnp.split(lstm_state, 2, axis=-1)
# Dense layer on the concatenation of x and h.
w, b = weights
y = jnp.dot(jnp.concatenate([x, h], axis=-1), w) + b
# i = input_gate, j = new_input, f = forget_gate, o = output_gate
i, j, f, o = jnp.split(y, 4, axis=-1)
new_c = c * math.sigmoid(f) + math.sigmoid(i) * jnp.tanh(j)
new_h = jnp.tanh(new_c) * math.sigmoid(o)
return new_h, jnp.concatenate([new_c, new_h], axis=-1)
def hash_vectors(self, vecs, rng):
# See https://arxiv.org/pdf/1509.02897.pdf
# We sample a different random rotation for each round of hashing to
# decrease the probability of hash misses.
if isinstance(self.n_buckets, int):
assert self.n_buckets % 2 == 0
rot_size = self.n_buckets
n_buckets = self.n_buckets
else:
# Factorize the hash if self.n_buckets is a list or tuple
rot_size, n_buckets = 0, 1
for factor in self.n_buckets:
assert factor % 2 == 0
rot_size += factor
n_buckets *= factor
rotations_shape = (vecs.shape[-1], self.n_hashes, rot_size // 2)
rng = jax.lax.stop_gradient(jax.lax.tie_in(vecs, rng))
random_rotations = jax.random.normal(rng,
rotations_shape).astype('float32')
rotated_vecs = np.einsum('tf,fhb->htb', vecs, random_rotations)
if isinstance(self.n_buckets, int) or len(self.n_buckets) == 1:
rotated_vecs = np.concatenate([rotated_vecs, -rotated_vecs],
axis=-1)
buckets = np.argmax(rotated_vecs, axis=-1)
else:
# Get the buckets for them and combine.
buckets, cur_sum, cur_product = None, 0, 1
for factor in self.n_buckets:
rv = rotated_vecs[..., cur_sum:cur_sum + (factor // 2)]
cur_sum += factor // 2
rv = np.concatenate([rv, -rv], axis=-1)
if buckets is None:
buckets = np.argmax(rv, axis=-1)
else:
buckets += cur_product * np.argmax(rv, axis=-1)
cur_product *= factor
# buckets is now (self.n_hashes, seqlen). Next we add offsets so that
# bucket numbers from different hashing rounds don't overlap.
offsets = jax.lax.tie_in(buckets, np.arange(self.n_hashes))
offsets = np.reshape(offsets * n_buckets, (-1, 1))
buckets = np.reshape(buckets + offsets, (-1, ))
return buckets
def forward(self, inp, weights):
"""Reshape input to have heads dimension and concatenate positions there."""
x = inp[0]
n_batches, seqlen = x.shape[0], x.shape[1]
d_head = x.shape[-1] // self._n_heads
res = np.reshape(x, (n_batches, seqlen, self._n_heads, d_head))
res = np.transpose(res, (0, 2, 1, 3)) # (batch, heads, len, depth)
if self._n_pos == 1: # Just one position given, tile into each head.
pos_shape = list(res.shape)[:-1] + [inp[1].shape[-1]]
pos = inp[1][:, None, :, :] + np.zeros(pos_shape) # Add 0 to broadcast.
else: # As many positions as heads, concatenate them in.
pos = [p[:, None, :, :] for p in inp[1:]]
pos = np.concatenate(pos, axis=1)
res = np.concatenate([res, pos], axis=-1)
# n_batch, n_heads, seqlen, d_head -> n_batch*n_heads, seqlen, d_head
res = np.reshape(res, (-1, seqlen, d_head + POS_VECTOR_SIZE))
return res
def forward(self, inputs, weights):
del weights
x1, x2 = inputs
x1_split = np.split(x1, self._n_sections, self._axis)
x2_split = np.split(x2, self._n_sections, self._axis)
res = [np.concatenate(ys, -1) for ys in zip(x1_split, x2_split)]
return tuple(res)
def _UpdateRow(x):
# row_e - (L1, H), row_d - (L2, H), row_mask_e - (L1,)
row_e, row_d, row_mask_e = x
# final_row - (L1+L2, H)
final_row = jnp.concatenate([row_e, jnp.zeros_like(row_d)], axis=0)
# Find the last real token/vector of the encoder.
e_idx = jnp.sum(row_mask_e, dtype=jnp.int32)
# Starting after that index, update with the decoder row.
return jax.lax.dynamic_update_slice(final_row, row_d, (e_idx, 0))
def test_fn_layer_weights_state(self):
layer = Fn('2in2out',
lambda x, y: (x + y, np.concatenate([x, y], axis=0)),
n_out=2)
input_signature = None
weights, state = layer.new_weights_and_state(input_signature)
self.assertIsNotNone(weights)
self.assertIsNotNone(state)
self.assertEmpty(weights)
self.assertEmpty(state)
def forward_with_state(self,
inputs,
weights=base.EMPTY_WEIGHTS,
state=base.EMPTY_STATE,
rng=None,
**kwargs):
embs = []
for ax_emb in weights:
ax_emb = np.broadcast_to(ax_emb, (inputs.shape[0], ) +
self._shape + (ax_emb.shape[-1], ))
embs.append(ax_emb)
if self._mode == 'predict':
assert self._dropout == 0.0
emb = np.concatenate(embs, -1)
emb = np.reshape(emb, (inputs.shape[0], -1, emb.shape[-1]))
emb = jax.lax.dynamic_slice_in_dim(emb,
state,
inputs.shape[1],
axis=1)
return inputs + emb, state + inputs.shape[1]
elif self._dropout == 0:
# TODO(kitaev): concat-then-reshape (as is the case with dropout enabled)
# leads to memory blow-up on TPU.
# emb = np.concatenate(embs, -1)
# return inputs + np.reshape(emb, inputs.shape), state
return inputs + np.concatenate([
np.reshape(emb, inputs.shape[:-1] + (emb.shape[-1], ))
for emb in embs
], -1), state
else:
emb = np.concatenate(embs, -1)
noise_shape = list(emb.shape)
for dim in self._dropout_broadcast_dims:
noise_shape[dim] = 1
keep_prob = 1.0 - self._dropout
if math.backend_name() == 'jax':
keep_prob = jax.lax.tie_in(
inputs, np.full((), keep_prob, dtype=inputs.dtype))
keep = math.random.bernoulli(rng, keep_prob, tuple(noise_shape))
multiplier = keep.astype(inputs.dtype) / keep_prob
return inputs + np.reshape(emb * multiplier, inputs.shape), state
def QueryPositionKV(x, keys=None, values=None, binary=False, **unused_kwargs):
"""Query a table with a position vector."""
if keys is None:
return x
k = np.array(keys)
v = np.array(values)
q = x
if binary:
q = np.concatenate([x, x], axis=-1)
return tl.DotProductAttention(q, k, v, None, 0.0, None, None)
def test_fn_layer_example(self):
layer = base.Fn(lambda x, y: (x + y, np.concatenate([x, y], axis=0)))
input_signature = (ShapeDtype((2, 7)), ShapeDtype((2, 7)))
expected_shape = ((2, 7), (4, 7))
output_shape = base.check_shape_agreement(layer, input_signature)
self.assertEqual(output_shape, expected_shape)
inp = (np.array([2]), np.array([3]))
x, xs = layer(inp)
self.assertEqual(int(x), 5)
self.assertEqual([int(y) for y in xs], [2, 3])
def test_fn_layer_difficult_n_out(self):
with self.assertRaisesRegex(ValueError, 'n_out'):
# Determining the output of this layer is hard with dummies.
base.Fn(lambda x: np.concatencate([x, x], axis=4))
# Check that this layer works when n_out is set.
layer = base.Fn(lambda x: np.concatenate([x, x], axis=4), n_out=1)
input_signature = ShapeDtype((2, 1, 2, 2, 3))
expected_shape = (2, 1, 2, 2, 6)
output_shape = base.check_shape_agreement(layer, input_signature)
self.assertEqual(output_shape, expected_shape)
def _get_embeddings(self, lo: int, hi: int, depth, rng=None):
"""Get embeddings float[length, depth].
Args:
lo: where to start sampling
hi: where to stop sampling
depth: embedding depth
rng: rng for random phase
Returns:
embeddings: float[length, depth]
"""
noise = self._get_noise(lo, hi, (depth + 1) // 2)
# Make the stddev around 1 after 1/drift.
noise = noise * self._drift**.5
t, c = onp.mgrid[lo:hi, :depth]
# Make even channels cos, odd channels sin:
c_div_2, c_mod_2 = divmod(c, 2)
# Off-by-one correction for odd depth:
drift = self._drift
if depth > 2:
drift = drift**(((depth + 1) // 2) / (depth // 2))
# Spend roughly half the frequencies on noise:
freq = np.geomspace(.5, .5 * drift**2, num=(depth + 1) // 2)[c_div_2]
cycles = c_mod_2 / 4 + freq * t + noise[:, c_div_2[0, :]] / 4
assert cycles.shape == (hi - lo, depth), cycles.shape
# Get random phases:
if self._affine:
assert rng is not None
cycles = cycles + trax.math.random.uniform(
rng, (
1,
depth,
), minval=0, maxval=1)
# Convert from cycles to radians:
embeddings = np.cos(np.pi * 2 * cycles)
# Set the last channels to the time bin features:
if self._time_bin_length is not None:
inter_bin_idx, intra_bin_idx = divmod(t[:, -1:],
self._time_bin_length)
bin_parity = inter_bin_idx % 2
bin_fraction = intra_bin_idx / self._time_bin_length
embeddings = np.concatenate([
embeddings[:, :-3],
1 / (1 + inter_bin_idx),
bin_fraction,
bin_parity.astype(np.float32),
], -1)
assert embeddings.shape == (hi - lo, depth), embeddings.shape
return embeddings
def look_adjacent(x, n_chunks_before, n_chunks_after):
"""Used to implement attention between consecutive chunks.
Args:
x: array of shape [n_chunks, chunk_len, ...]
n_chunks_before: Number of previous chunks to attend to.
n_chunks_after: Number of subsequent chunks to attend to.
Returns:
array of shape [n_chunks, N * chunk_len, ...], where
N = (1 + n_chunks_before + n_chunks_after).
"""
if n_chunks_before == 0 and n_chunks_after == 0:
return x
slices = []
for i in range(-n_chunks_before, n_chunks_after + 1):
if i == 0:
slices.append(x)
else:
slices.append(np.concatenate([x[i:, ...], x[:i, ...]], axis=0))
return np.concatenate(slices, axis=1)
def f(x): # pylint: disable=invalid-name
# x : [batch, 1, length, depth]
x = np.pad(x, [(0, 0), (0, 0), (1, 1), (0, 0)],
mode='constant',
constant_values=0.0)
depth = x.shape[-1] // 3
assert 3 * depth == x.shape[-1], ('Depth must be divisible by 3',
depth, x.shape)
xs = [
x[:, :, :-2, :depth], x[:, :, 1:-1, depth:2 * depth],
x[:, :, 2:, 2 * depth:3 * depth]
]
return np.concatenate(xs, axis=3)
def DiagonalGate(x):
"""Split channels in 3 parts. Shifts 1st and 3rd sections to left/right."""
# x : [batch, 1, length, depth]
x = np.pad(
x, [(0, 0), (0, 0), (1, 1), (0, 0)], mode='constant', constant_values=0.0)
depth = x.shape[-1] // 3
assert 3 * depth == x.shape[-1], ('Depth must be divisible by 3', depth,
x.shape)
xs = [
x[:, :, :-2, :depth], x[:, :, 1:-1, depth:2 * depth],
x[:, :, 2:, 2 * depth:3 * depth]
]
return np.concatenate(xs, axis=3)
def Deinterleave(inputs, x_size, y_size, **unused_kwargs):
"""Inverse of Interleave."""
reprs = inputs
(batch_size, length) = reprs.shape[:2]
shape_suffix = reprs.shape[2:]
remainder_length = length % (x_size + y_size)
remainder = reprs[:, None, -remainder_length:]
reprs = reprs[:, :-remainder_length]
reprs = np.reshape(reprs, (batch_size, -1, x_size + y_size) + shape_suffix)
x_reprs = reprs[:, :, :x_size]
y_reprs = reprs[:, :, x_size:]
x_reprs = np.concatenate((x_reprs, remainder), axis=1)
return (x_reprs, y_reprs)
def F(x):
# TODO(afrozm): What to do in this case?
if mode == 'predict':
raise ValueError('MaskOfRightShiftedArray not implemented for predict.')
mask = x != 0
if n_shifts == 0:
return mask
# Need to set (B, n_shifts, ...) section to True.
trues_shape = (x.shape[0], n_shifts) + mask.shape[2:]
trues = jnp.full(trues_shape, True)
return jnp.concatenate([trues, mask[:, n_shifts:, ...]], axis=1)
def F(vec_e, vec_d, mask_e, mask_d):
# pylint: disable=invalid-name
L1 = mask_e.shape[1]
L2 = mask_d.shape[1]
# pylint: enable=invalid-name
# [-(L1+L2), -L2) but with padding 0-ed out - (B, L1).
mask_e_key = jnp.arange(-(L1 + L2), -L2) * mask_e
# [-L2,0) but with padding 0-ed out - (B, L2).
mask_d_key = jnp.arange(-L2, 0) * mask_d
# Shape (B, L1+L2, H)
enc_dec_concat = jnp.concatenate([vec_e, vec_d], axis=1)
# Shape (B, L1+L2)
mask_concat = jnp.concatenate([mask_e_key, mask_d_key], axis=1)
# Make `mask_concat` the same shape as `enc_dec_concat`
mask_concat = (
mask_concat[..., jnp.newaxis] +
jnp.zeros_like(enc_dec_concat, dtype=jnp.int32))
# Sort on `mask_concat` so padding with key=0 goes to the right end, axis=1.
_, enc_dec_pad = math.sort_key_val(mask_concat, enc_dec_concat, 1)
return enc_dec_pad
def deinterleave(inputs):
reprs = inputs
(batch_size, length) = reprs.shape[:2]
shape_suffix = reprs.shape[2:]
remainder_length = length % (x_size + y_size)
if remainder_length > 0:
remainder = reprs[:, None, -remainder_length:]
reprs = reprs[:, :-remainder_length]
reprs = jnp.reshape(reprs,
(batch_size, -1, x_size + y_size) + shape_suffix)
x_reprs = reprs[:, :, :x_size]
y_reprs = reprs[:, :, x_size:]
if remainder_length > 0:
x_reprs = jnp.concatenate((x_reprs, remainder), axis=1)
return (x_reprs, y_reprs)
