diff --git a/src/moscot/problems/_utils.py b/src/moscot/problems/_utils.py
index 64ce1dc24..8417e1c9a 100644
--- a/src/moscot/problems/_utils.py
+++ b/src/moscot/problems/_utils.py
@@ -82,11 +82,11 @@ def handle_joint_attr(
             "y_key": joint_attr,
         }
         return xy, kwargs
-    if isinstance(joint_attr, Mapping):
+    if isinstance(joint_attr, Mapping):  # input mapping does not distinguish between x and y as it's a shared space
         joint_attr = dict(joint_attr)
-        if "attr" in joint_attr and joint_attr["attr"] == "X":
+        if "attr" in joint_attr and joint_attr["attr"] == "X":  # we have a point cloud
             return {"x_attr": "X", "y_attr": "X"}, kwargs
-        if "attr" in joint_attr and joint_attr["attr"] == "obsm":
+        if "attr" in joint_attr and joint_attr["attr"] == "obsm":  # we have a point cloud
             if "key" not in joint_attr:
                 raise KeyError("`key` must be provided when `attr` is `obsm`.")
             xy = {
@@ -96,8 +96,8 @@ def handle_joint_attr(
                 "y_key": joint_attr["key"],
             }
             return xy, kwargs
-        if joint_attr.get("tag", None) == "cost_matrix":
-            if len(joint_attr) == 2 or kwargs.get("attr", None) == "obsp":
+        if joint_attr.get("tag", None) == "cost_matrix":  # if this is True we have custom cost matrix or moscot cost
+            if len(joint_attr) == 2 or kwargs.get("attr", None) == "obsp":  # in this case we have a custom cost matrix
                 joint_attr.setdefault("cost", "custom")
                 joint_attr.setdefault("attr", "obsp")
                 kwargs["xy_callback"] = "cost-matrix"
diff --git a/src/moscot/problems/base/_base_problem.py b/src/moscot/problems/base/_base_problem.py
index e58a682b3..cb77dcd22 100644
--- a/src/moscot/problems/base/_base_problem.py
+++ b/src/moscot/problems/base/_base_problem.py
@@ -458,7 +458,7 @@ def concat(x: ArrayLike, y: ArrayLike) -> ArrayLike:
             if scaler is not None:
                 data = scaler.fit_transform(data)
             return {"xy": TaggedArray(data[:n], data[n:], tag=Tag.POINT_CLOUD)}
-        if term in ("x", "y"):
+        if term in ("x", "y"):  # if we don't have a shared space, then adata_y is always None
             logger.info(f"Computing pca with `n_comps={n_comps}` for `{term}` using `{msg}`")
             x = sc.pp.pca(x, n_comps=n_comps, **kwargs)
             if scaler is not None:
diff --git a/src/moscot/problems/base/_compound_problem.py b/src/moscot/problems/base/_compound_problem.py
index d1ec3f5d4..8b8ca61ca 100644
--- a/src/moscot/problems/base/_compound_problem.py
+++ b/src/moscot/problems/base/_compound_problem.py
@@ -104,7 +104,7 @@ def _callback_handler(
         key_2: K,
         problem: B,
         *,
-        callback: Union[Literal["local-pca"], Callback_t],
+        callback: Optional[Union[Literal["local-pca"], Callback_t]] = None,
         **kwargs: Any,
     ) -> Mapping[Literal["xy", "x", "y"], TaggedArray]:
         def verify_data(data: Mapping[Literal["xy", "x", "y"], TaggedArray]) -> None:
@@ -115,6 +115,8 @@ def verify_data(data: Mapping[Literal["xy", "x", "y"], TaggedArray]) -> None:
                 if not isinstance(val, TaggedArray):
                     raise TypeError(f"Expected value for `{key}` to be a `TaggedArray`, found `{type(val)}`.")
 
+        if callback is None:
+            return {}
         if callback == "local-pca":
             callback = problem._local_pca_callback
 
@@ -150,24 +152,19 @@ def _create_problems(
                 tgt_name = tgt
 
             problem = self._create_problem(src, tgt, src_mask=src_mask, tgt_mask=tgt_mask)
-            if xy_callback is not None:
-                xy_data = self._callback_handler(
-                    term="xy", key_1=src, key_2=tgt, problem=problem, callback=xy_callback, **xy_callback_kwargs
-                )
-            else:
-                xy_data = {}
-            if x_callback is not None:
-                x_data = self._callback_handler(
-                    term="x", key_1=src, key_2=tgt, problem=problem, callback=x_callback, **x_callback_kwargs
-                )
-            else:
-                x_data = {}
-            if y_callback is not None:
-                y_data = self._callback_handler(
-                    term="y", key_1=src, key_2=tgt, problem=problem, callback=y_callback, **y_callback_kwargs
-                )
-            else:
-                y_data = {}
+
+            xy_data = self._callback_handler(
+                term="xy", key_1=src, key_2=tgt, problem=problem, callback=xy_callback, **xy_callback_kwargs
+            )
+
+            x_data = self._callback_handler(
+                term="x", key_1=src, key_2=tgt, problem=problem, callback=x_callback, **x_callback_kwargs
+            )
+
+            y_data = self._callback_handler(
+                term="y", key_1=src, key_2=tgt, problem=problem, callback=y_callback, **y_callback_kwargs
+            )
+
             kws = {**kwargs, **xy_data, **x_data, **y_data}  # type: ignore[arg-type]
 
             if isinstance(problem, BirthDeathProblem):
@@ -626,7 +623,7 @@ def _callback_handler(
         key_2: K,
         problem: B,
         *,
-        callback: Union[Literal["local-pca", "cost-matrix"], Callback_t],
+        callback: Optional[Union[Literal["local-pca", "cost-matrix"], Callback_t]] = None,
         **kwargs: Any,
     ) -> Mapping[Literal["xy", "x", "y"], TaggedArray]:
         # TODO(michalk8): better name?
@@ -657,10 +654,15 @@ def _cost_matrix_callback(
 
             linear_cost_matrix = data[mask, :][:, mask_2]
             if issparse(linear_cost_matrix):
+                logger.warning("Linear cost matrix being densified.")
                 linear_cost_matrix = linear_cost_matrix.A
             return {"xy": TaggedArray(linear_cost_matrix, tag=Tag.COST_MATRIX)}
 
         if term in ("x", "y"):
-            return {term: TaggedArray(data[mask, :][:, mask], tag=Tag.COST_MATRIX)}
+            quad_cost_matrix = data[mask, :][:, mask_2]
+            if issparse(quad_cost_matrix):
+                logger.warning("Quadratic cost matrix being densified.")
+                quad_cost_matrix = quad_cost_matrix.A
+            return {term: TaggedArray(quad_cost_matrix, tag=Tag.COST_MATRIX)}
 
         raise ValueError(f"Expected `term` to be one of `x`, `y`, or `xy`, found `{term!r}`.")