diff --git a/lib/polaris/updates.ex b/lib/polaris/updates.ex
index b259a4a..91829d4 100644
--- a/lib/polaris/updates.ex
+++ b/lib/polaris/updates.ex
@@ -1003,6 +1003,64 @@ defmodule Polaris.Updates do
     Map.merge(params, state, fn _, s1, s2 -> merge_inner(s1, s2) end)
   end
 
+  @doc """
+  Applies the GaLore algorithm to an optimizer for low-memory
+  training.
+  """
+  def galore({parent_init_fn, parent_apply_fn}, galore_params, opts \\ []) do
+    opts = Keyword.validate!(opts, rank: 128, scale: 1.0)
+
+    init_fn = fn params ->
+      # on initialization, project down so we initialize parent
+      # state with low-rank version
+      {galore, regular} = Map.split(params, galore_params)
+      {projected, _ortho_matrix} = apply_galore_projection_down(galore, opts)
+      parent_init_fn.(Map.merge(projected, regular))
+    end
+
+    apply_fn = fn updates, state, params ->
+      {galore, regular} = Map.split(updates, galore_params)
+      {projected, ortho_matrix} = apply_galore_projection_down(galore, opts)
+      {scaled_updates, new_state} = parent_apply_fn.(Map.merge(projected, regular), state, params)
+      {galore, regular} = Map.split(scaled_updates, galore_params)
+      galore_updates = apply_galore_projection_up(galore, ortho_matrix, opts)
+      {Map.merge(galore_updates, regular), new_state}
+    end
+
+    {init_fn, apply_fn}
+  end
+
+  defnp apply_galore_projection_down(params, opts \\ []) do
+    opts = keyword!(opts, rank: 128, scale: 1.0)
+
+    ortho_matrix =
+      deep_new(params, fn g ->
+        get_orthogonal_matrix(g, rank: opts[:rank])
+      end)
+
+    projected =
+      deep_merge(params, ortho_matrix, fn g, ortho ->
+        Nx.dot(g, Nx.transpose(ortho))
+      end)
+
+    {projected, ortho_matrix}
+  end
+
+  defnp apply_galore_projection_up(params, ortho_matrix, opts \\ []) do
+    opts = keyword!(opts, rank: 128, scale: 1.0)
+
+    deep_merge(params, ortho_matrix, fn g, ortho ->
+      opts[:scale] * Nx.dot(g, ortho)
+    end)
+  end
+
+  defnp get_orthogonal_matrix(g, opts \\ []) do
+    opts = keyword!(opts, rank: 128)
+
+    {_u, _s, vh} = Nx.LinAlg.svd(g, full_matrices?: false)
+    vh[[0..opts[:rank], ..]]
+  end
+
   ## Helpers
 
   defnp update_moment(x, moment, decay, order) do