diff --git a/src/compressed_tensors/compressors/model_compressors/model_compressor.py b/src/compressed_tensors/compressors/model_compressors/model_compressor.py
index 700c1769..6bd5421b 100644
--- a/src/compressed_tensors/compressors/model_compressors/model_compressor.py
+++ b/src/compressed_tensors/compressors/model_compressors/model_compressor.py
@@ -479,7 +479,7 @@ def decompress_model(self, model: Module):
                 # remove any existing parameters
                 exec_device = get_execution_device(module)
                 offload_device = get_offloaded_device(module)
-                for name, _ in list(module.named_parameters()):
+                for name, _ in list(module.named_parameters(recurse=False)):
                     delete_offload_parameter(module, name)
 
                 # replace with decompressed parameters
diff --git a/src/compressed_tensors/transform/factory/base.py b/src/compressed_tensors/transform/factory/base.py
index 4c7b6c91..9db318a0 100644
--- a/src/compressed_tensors/transform/factory/base.py
+++ b/src/compressed_tensors/transform/factory/base.py
@@ -26,6 +26,7 @@
 )
 from compressed_tensors.utils import (
     align_module_device,
+    delete_offload_module,
     has_offloaded_params,
     patch_attr,
     register_offload_module,
@@ -99,7 +100,7 @@ def _apply_to_module(self, module: Module, args: TransformArgs):
         # create transform as submodule
         transform_name = f"{self.name}_{args.location}"
         transform = self.create_transform(module, args)
-        register_offload_module(module, transform_name, transform)  # (1)
+        register_offload_module(module, transform_name, transform)
 
         # register input transformation hook
         if args.location == TransformLocation.INPUT:
@@ -128,6 +129,10 @@ def input_hook(_, args):
                     raise ValueError("Offloaded training is not supported")
                 P.register_parametrization(module, "weight", transform)
 
+            else:
+                # if we're not training, there's no reason to keep the transform
+                delete_offload_module(module, transform_name)
+
         # register output transformation hook
         elif args.location == TransformLocation.OUTPUT:
 
@@ -140,9 +145,6 @@ def output_hook(_, _input, output):
         else:
             raise NotImplementedError()
 
-        # (1) even in the `weight` cases, this submodule attachment is needed in order
-        # to support saving in the frozen state
-
 
 class TransformBase(Module, ABC):
     """
diff --git a/src/compressed_tensors/transform/factory/hadamard.py b/src/compressed_tensors/transform/factory/hadamard.py
index b1da88a3..b4d5f7de 100644
--- a/src/compressed_tensors/transform/factory/hadamard.py
+++ b/src/compressed_tensors/transform/factory/hadamard.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import Optional
+from typing import Optional, Union
 
 import torch
 from compressed_tensors.transform import TransformArgs, TransformScheme
@@ -41,6 +41,7 @@ class HadamardFactory(TransformFactory):
     def __init__(self, name: str, scheme: TransformScheme, seed: Optional[int] = None):
         super().__init__(name, scheme, seed)
         self.weights = ParameterizedDefaultDict(self._create_weight)
+        self.perms = ParameterizedDefaultDict(self._create_permutation)
 
     def create_transform(self, module: Module, args: TransformArgs):
         """
@@ -56,24 +57,35 @@ def create_transform(self, module: Module, args: TransformArgs):
         device = get_offloaded_device(module)
 
         weight = self.weights[size, dtype, device]
-        return HadamardTransform(weight, args)
+        perm = self.perms[weight] if self.scheme.randomize else None
+        return HadamardTransform(weight, perm, args)
 
     def _create_weight(self, size: int, dtype: dtype, device: device) -> Parameter:
         data = deterministic_hadamard_matrix(size)
         data = data.to(dtype=dtype, device=device)
         return Parameter(data, requires_grad=self.scheme.requires_grad)
 
+    def _create_permutation(self, weight: Parameter) -> Parameter:
+        data = torch.randperm(weight.size(0), generator=self.generator)
+        return Parameter(data, requires_grad=False)
+
 
 class HadamardTransform(TransformBase):
-    def __init__(self, weight: Parameter, args: TransformArgs):
+    def __init__(
+        self, weight: Parameter, perm: Union[Parameter, None], args: TransformArgs
+    ):
         super().__init__()
         self.weight = weight
+        self.perm = perm
         self.args = args
 
     def forward(self, value: Tensor) -> Tensor:
-        if not self.args.inverse:
-            weight = self.weight
-        else:
-            weight = self.weight.T
+        weight = self.weight
+
+        if self.perm is not None:
+            weight = weight[self.perm][:, self.perm]
+
+        if self.args.inverse:
+            weight = weight.T
 
         return apply_transform_weight(weight, value, self.args.location)
diff --git a/src/compressed_tensors/transform/transform_config.py b/src/compressed_tensors/transform/transform_config.py
index 414c21e0..df178c42 100644
--- a/src/compressed_tensors/transform/transform_config.py
+++ b/src/compressed_tensors/transform/transform_config.py
@@ -49,7 +49,7 @@ class TransformConfig(BaseModel):
                     inverse=True,
                 ),
             ],
-            randomize_modules=True,
+            randomize=True,
         ),
         "u": TransformScheme(
             type="hadamard",
@@ -62,7 +62,7 @@ class TransformConfig(BaseModel):
                     targets=["Linear"], location="output", inverse=True  # non-mergable
                 ),
             ],
-            randomize_modules=True,
+            randomize=True,
         ),
     }
 )
diff --git a/src/compressed_tensors/transform/transform_scheme.py b/src/compressed_tensors/transform/transform_scheme.py
index 1335063c..64d646e0 100644
--- a/src/compressed_tensors/transform/transform_scheme.py
+++ b/src/compressed_tensors/transform/transform_scheme.py
@@ -31,13 +31,12 @@ class TransformScheme(BaseModel):
         (see `Transforms.registered_names()`)
     :param apply: list of TransformationArgs containing the information about the
         modules that should be targeted by the specified transform
-    :param randomize_modules: True if unique transforms should be applied to each
-        unique module targeted by `apply`, otherwise reuse transform weights where
-        applicable
+    :param randomize: True if uniquely randomized transform weights should be used,
+        otherwise use identical transform weights where applicable
     :param requires_grad: True if weights include gradients for training
     """
 
     type: str
     apply: List[TransformArgs] = Field(default_factory=list)
-    randomize_modules: bool = Field(default=False)
+    randomize: bool = Field(default=False)
     requires_grad: bool = Field(default=False)
diff --git a/src/compressed_tensors/utils/offload.py b/src/compressed_tensors/utils/offload.py
index c1a46e3c..7cc9f33c 100644
--- a/src/compressed_tensors/utils/offload.py
+++ b/src/compressed_tensors/utils/offload.py
@@ -467,6 +467,7 @@ def register_offload_module(base: torch.nn.Module, name: str, module: torch.nn.M
 def delete_offload_module(base: torch.nn.Module, name: str):
     """
     Delete a submodule from a model which may contain offloading
+
     :param base: parent module to delete submodule from
     :param name: name of submodule on parent
     """
diff --git a/tests/test_transform/factory/test_compression.py b/tests/test_transform/factory/test_compression.py
new file mode 100644
index 00000000..6f763fcf
--- /dev/null
+++ b/tests/test_transform/factory/test_compression.py
@@ -0,0 +1,126 @@
+# Copyright (c) 2021 - present / Neuralmagic, Inc. All Rights Reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#    http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import torch
+from compressed_tensors import ModelCompressor
+from compressed_tensors.quantization import QuantizationStatus
+from compressed_tensors.transform import (
+    TransformArgs,
+    TransformBase,
+    TransformFactory,
+    TransformLocation,
+    TransformScheme,
+)
+
+
+class TransformableModel(torch.nn.Module):
+    def __init__(self, *sizes):
+        super().__init__()
+        self.fcs = torch.nn.ModuleList([])
+        self.fcs.append(torch.nn.Linear(sizes[0], sizes[1], bias=False))
+        for index in range(1, len(sizes) - 1):
+            self.fcs.append(torch.nn.Linear(sizes[index], sizes[index + 1], bias=False))
+
+    def forward(self, x):
+        for layer in self.fcs:
+            x = layer(x)
+        return x
+
+
+def test_frozen_reload():
+    # TODO: test applying and reloadaing a transformers model
+    pass
+
+
+def test_compressed_keys():
+    model = TransformableModel(2, 4, 8, 16, 32, 64)
+
+    scheme = TransformScheme(type="hadamard")
+    scheme.apply = [
+        TransformArgs(targets="fcs.0", location="weight_output"),
+        TransformArgs(targets="fcs.1", location="input", inverse=True),
+        TransformArgs(targets="fcs.1", location="output"),
+        TransformArgs(targets="fcs.2", location="weight_input", inverse=True),
+        TransformArgs(targets="fcs.2", location="output"),
+        TransformArgs(targets="fcs.3", location="input", inverse=True),
+        TransformArgs(targets="fcs.3", location="weight_output"),
+        TransformArgs(targets="fcs.4", location="weight_input", inverse=True),
+    ]
+    factory = TransformFactory.from_scheme(scheme, name="")
+
+    input = torch.rand((17, model.fcs[0].in_features))
+    true_output = model(input)
+
+    factory.apply_to_model(model)
+
+    compressor = ModelCompressor()
+    compressor.compress_model(model)
+
+    keys = {
+        "fcs.0.weight",
+        "fcs.1.weight",
+        "fcs.1._input.weight",
+        "fcs.1._output.weight",
+        "fcs.2.weight",
+        "fcs.2._output.weight",
+        "fcs.3.weight",
+        "fcs.3._input.weight",
+        "fcs.4.weight",
+    }
+    assert model.state_dict().keys() == keys
+
+    output = model(input)
+    assert torch.allclose(true_output, output, atol=1e-7, rtol=0.0)
+
+
+def test_compress_decompress():
+    model = TransformableModel(2, 4, 8, 16, 32, 64)
+
+    scheme = TransformScheme(type="hadamard")
+    scheme.apply = [
+        TransformArgs(targets="fcs.0", location="weight_output"),
+        TransformArgs(targets="fcs.1", location="input", inverse=True),
+        TransformArgs(targets="fcs.1", location="output"),
+        TransformArgs(targets="fcs.2", location="weight_input", inverse=True),
+        TransformArgs(targets="fcs.2", location="output"),
+        TransformArgs(targets="fcs.3", location="input", inverse=True),
+        TransformArgs(targets="fcs.3", location="weight_output"),
+        TransformArgs(targets="fcs.4", location="weight_input", inverse=True),
+    ]
+    factory = TransformFactory.from_scheme(scheme, name="")
+
+    input = torch.rand((17, model.fcs[0].in_features))
+    true_output = model(input)
+
+    factory.apply_to_model(model)
+
+    compressor = ModelCompressor()
+    compressor.compress_model(model)
+    compressor.decompress_model(model)
+
+    output = model(input)
+    assert torch.allclose(true_output, output, atol=1e-7, rtol=0.0)
+
+    keys = {
+        "fcs.0.weight",
+        "fcs.1.weight",
+        "fcs.1._input.weight",
+        "fcs.1._output.weight",
+        "fcs.2.weight",
+        "fcs.2._output.weight",
+        "fcs.3.weight",
+        "fcs.3._input.weight",
+        "fcs.4.weight",
+    }
+    assert model.state_dict().keys() == keys
diff --git a/tests/test_transform/factory/test_correctness.py b/tests/test_transform/factory/test_correctness.py
index bab1117e..1745281f 100644
--- a/tests/test_transform/factory/test_correctness.py
+++ b/tests/test_transform/factory/test_correctness.py
@@ -19,10 +19,17 @@
     TransformFactory,
     TransformScheme,
 )
-from compressed_tensors.utils import align_modules, force_cpu_offload
+from compressed_tensors.utils import force_cpu_offload
 from tests.testing_utils import requires_accelerate, requires_gpu
 
 
+def all_schemes():
+    all_types = TransformFactory.registered_names()
+    base = [TransformScheme(type=type) for type in all_types]
+    randomized = [TransformScheme(type=type, randomize=True) for type in all_types]
+    return base + randomized
+
+
 class TransformableModel(torch.nn.Module):
     def __init__(self, *sizes):
         super().__init__()
@@ -37,10 +44,7 @@ def forward(self, x):
         return x
 
 
-@pytest.mark.parametrize(
-    "scheme",
-    [TransformScheme(type=name) for name in TransformFactory.registered_names()],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_correctness_linear(scheme):
     size = (4, 8)
     module = torch.nn.Linear(*size, bias=True)
@@ -67,10 +71,7 @@ def test_correctness_linear(scheme):
     assert torch.allclose(true_output, output, atol=1e-5, rtol=0.0)
 
 
-@pytest.mark.parametrize(
-    "scheme",
-    [TransformScheme(type=name) for name in TransformFactory.registered_names()],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_correctness_model(scheme, offload=False):
     # load model
     model = TransformableModel(2, 4, 8, 16, 32, 64)
@@ -108,9 +109,6 @@ def test_correctness_model(scheme, offload=False):
 
 @requires_gpu
 @requires_accelerate()
-@pytest.mark.parametrize(
-    "scheme",
-    [TransformScheme(type=name) for name in TransformFactory.registered_names()],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_correctness_model_offload(scheme):
     test_correctness_model(scheme, offload=True)
diff --git a/tests/test_transform/factory/test_memory.py b/tests/test_transform/factory/test_memory.py
index 49e882e4..15d72b9b 100644
--- a/tests/test_transform/factory/test_memory.py
+++ b/tests/test_transform/factory/test_memory.py
@@ -26,6 +26,13 @@
 from tests.testing_utils import requires_accelerate, requires_gpu
 
 
+def all_schemes():
+    all_types = TransformFactory.registered_names()
+    base = [TransformScheme(type=type) for type in all_types]
+    randomized = [TransformScheme(type=type, randomize=True) for type in all_types]
+    return base + randomized
+
+
 class TransformableModel(torch.nn.Module):
     def __init__(self, *sizes):
         super().__init__()
@@ -40,10 +47,7 @@ def forward(self, x):
         return x
 
 
-@pytest.mark.parametrize(
-    "scheme",
-    [TransformScheme(type=name) for name in TransformFactory.registered_names()],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_memory_sharing(scheme, offload=False):
     # load scheme and factory
     scheme = TransformScheme(
@@ -93,20 +97,12 @@ def test_memory_sharing(scheme, offload=False):
 
 @requires_gpu
 @requires_accelerate()
-@pytest.mark.parametrize(
-    "scheme",
-    [TransformScheme(type=name) for name in TransformFactory.registered_names()],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_memory_sharing_offload(scheme):
     test_memory_sharing(scheme, offload=True)
 
 
-@pytest.mark.parametrize(
-    "scheme",
-    [
-        TransformScheme(type=name, requires_grad=True)
-        for name in TransformFactory.registered_names()
-    ],
-)
+@pytest.mark.parametrize("scheme", all_schemes())
 def test_memory_sharing_training(scheme):
+    scheme.requires_grad = True
     test_memory_sharing(scheme, offload=False)
diff --git a/tests/test_transform/test_transform_scheme.py b/tests/test_transform/test_transform_scheme.py
index ad851762..839ab46a 100644
--- a/tests/test_transform/test_transform_scheme.py
+++ b/tests/test_transform/test_transform_scheme.py
@@ -24,7 +24,7 @@ def test_basic_scheme():
         type="hadamard",
         apply=[basic_args],
     )
-    assert not scheme.randomize_modules
+    assert not scheme.randomize
     assert scheme.type == "hadamard"
     assert len(scheme.apply) == 1
     assert isinstance(scheme.apply[0], TransformArgs)
@@ -43,10 +43,10 @@ def test_multiple_groups_global():
     scheme = TransformScheme(
         type="hadamard",
         apply=[embedding_args, linear_args],
-        randomize_modules=True,
+        randomize=True,
     )
 
-    assert scheme.randomize_modules
+    assert scheme.randomize
     assert scheme.type == "hadamard"
     assert len(scheme.apply) == 2
     assert isinstance(scheme.apply[0], TransformArgs)
@@ -69,6 +69,6 @@ def test_multiple_groups():
         apply=apply,
     )
 
-    assert not scheme.randomize_modules
+    assert not scheme.randomize
     assert scheme.type == "hadamard"
     assert len(scheme.apply) == 20