diff --git a/src/ott/neural/methods/flows/genot.py b/src/ott/neural/methods/flows/genot.py
index 7bc989fd7..d0d88fd04 100644
--- a/src/ott/neural/methods/flows/genot.py
+++ b/src/ott/neural/methods/flows/genot.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 import functools
-from typing import Any, Callable, Dict, Iterable, List, Optional, Tuple
+from typing import Any, Callable, Dict, Iterable, List, Optional, Tuple, Union
 
 import jax
 import jax.numpy as jnp
@@ -29,12 +29,11 @@
 
 __all__ = ["GENOT"]
 
-# input: (src_lin, tgt_lin, src_quad, tgt_quad), output: (len(src), len(tgt))
-# all are optional because the problem can be linear/quadratic/fused
-DataMatchFn_t = Callable[[
-    Optional[jnp.ndarray], Optional[jnp.ndarray], Optional[jnp.ndarray],
-    Optional[jnp.ndarray]
-], jnp.ndarray]
+LinTerm = Tuple[jnp.ndarray, jnp.ndarray]
+QuadTerm = Tuple[jnp.ndarray, jnp.ndarray, Optional[jnp.ndarray],
+                 Optional[jnp.ndarray]]
+DataMatchFn = Union[Callable[[LinTerm], jnp.ndarray], Callable[[QuadTerm],
+                                                               jnp.ndarray]]
 
 
 class GENOT:
@@ -49,8 +48,14 @@ class GENOT:
     vf: Vector field parameterized by a neural network.
     flow: Flow between the latent and the target distributions.
     data_match_fn: Function to match samples from the source and the target
-      distributions with a ``(src_lin, tgt_lin, src_quad, tgt_quad) -> matching``
-      signature.
+      distributions. Depending on the data passed in :meth:`__call__`, it has
+      the following signature:
+
+      - ``(src_lin, tgt_lin) -> matching`` - linear matching.
+      - ``(src_quad, tgt_quad, src_lin, tgt_lin) -> matching`` -
+        quadratic (fused) GW matching. In the pure GW setting, btoh ``src_lin``
+        and ``tgt_lin`` will be set to :obj:`None`.
+
     source_dim: Dimensionality of the source distribution.
     target_dim: Dimensionality of the target distribution.
     condition_dim: Dimension of the conditions. If :obj:`None`, the underlying
@@ -73,7 +78,7 @@ def __init__(
       self,
       vf: velocity_field.VelocityField,
       flow: dynamics.BaseFlow,
-      data_match_fn: DataMatchFn_t,
+      data_match_fn: DataMatchFn,
       *,
       source_dim: int,
       target_dim: int,
@@ -167,16 +172,19 @@ def prepare_data(
                      Optional[jnp.ndarray]]]:
       src_lin, src_quad = batch.get("src_lin"), batch.get("src_quad")
       tgt_lin, tgt_quad = batch.get("tgt_lin"), batch.get("tgt_quad")
-      arrs = src_lin, tgt_lin, src_quad, tgt_quad
 
       if src_quad is None and tgt_quad is None:  # lin
         src, tgt = src_lin, tgt_lin
-        arrs = src_lin, tgt_lin  # get rid of src_quad, tgt_quad args
+        arrs = src_lin, tgt_lin
       elif src_lin is None and tgt_lin is None:  # quad
         src, tgt = src_quad, tgt_quad
-      elif all(arr is not None for arr in arrs):  # fused quad
+        arrs = src_quad, tgt_quad
+      elif all(
+          arr is not None for arr in (src_lin, tgt_lin, src_quad, tgt_quad)
+      ):  # fused quad
         src = jnp.concatenate([src_lin, src_quad], axis=1)
         tgt = jnp.concatenate([tgt_lin, tgt_quad], axis=1)
+        arrs = src_quad, tgt_quad, src_lin, tgt_lin
       else:
         raise RuntimeError("Cannot infer OT problem type from data.")
 
diff --git a/src/ott/solvers/utils.py b/src/ott/solvers/utils.py
index 667d74bf5..f7bdae63a 100644
--- a/src/ott/solvers/utils.py
+++ b/src/ott/solvers/utils.py
@@ -59,10 +59,10 @@ def match_linear(
 
 
 def match_quadratic(
-    x: Optional[jnp.ndarray],
-    y: Optional[jnp.ndarray],
     xx: jnp.ndarray,
     yy: jnp.ndarray,
+    x: Optional[jnp.ndarray] = None,
+    y: Optional[jnp.ndarray] = None,
     scale_cost: ScaleCost_t = 1.0,
     cost_fn: Optional[costs.CostFn] = None,
     **kwargs: Any
diff --git a/tests/neural/methods/genot_test.py b/tests/neural/methods/genot_test.py
index 78f0aa8cb..d4d8a1399 100644
--- a/tests/neural/methods/genot_test.py
+++ b/tests/neural/methods/genot_test.py
@@ -32,8 +32,9 @@ def get_match_fn(typ: Literal["lin", "quad", "fused"]):
     return solver_utils.match_linear
   if typ == "quad":
     return solver_utils.match_quadratic
-  # typ == "fused"
-  return solver_utils.match_quadratic
+  if typ == "fused":
+    return solver_utils.match_quadratic
+  raise NotImplementedError(typ)
 
 
 class TestGENOT:
@@ -61,7 +62,7 @@ def test_genot(self, rng: jax.Array, dl: str, request):
     model = genot.GENOT(
         vf,
         flow=dynamics.ConstantNoiseFlow(0.0),
-        data_match_fn=get_match_fn(typ=problem_type),
+        data_match_fn=get_match_fn(problem_type),
         source_dim=src_dim,
         target_dim=tgt_dim,
         condition_dim=cond_dim,