diff --git a/src/autoschedulers/mullapudi2016/AutoSchedule.cpp b/src/autoschedulers/mullapudi2016/AutoSchedule.cpp
index 21c9f1268351..6304ada57958 100644
--- a/src/autoschedulers/mullapudi2016/AutoSchedule.cpp
+++ b/src/autoschedulers/mullapudi2016/AutoSchedule.cpp
@@ -945,10 +945,10 @@ class GPUTileHelper {
             return;
         }
 
-        internal_assert(vars.size() <= 3);
-
         std::stringstream oss;
         switch (vars.size()) {
+        case 0:
+            return;
         case 1: {
             const auto &[v, outer, inner, factor, strategy] = vars.front();
             f.split(v, outer, inner, factor, strategy);
@@ -996,7 +996,7 @@ class GPUTileHelper {
 
             break;
         }
-        case 3: {
+        default: {
             const auto &x = vars[0];
             const auto &y = vars[1];
             const auto &z = vars[2];
@@ -1086,6 +1086,7 @@ class GPUTilingDedup {
     bool is_initial_order = true;
     std::vector<VarOrRVar> ordering;
 
+    std::set<std::string> is_split;
     std::set<std::string> outer_vars;
     std::set<std::string> inner_vars;
 
@@ -1099,6 +1100,11 @@ class GPUTilingDedup {
         return inner_vars.find(variable_name) != inner_vars.end();
     }
 
+    /** True if Func::parallel(v_o) is pending. */
+    bool is_parallel(const std::string &variable_name) const {
+        return parallelize.find(variable_name) != parallelize.end();
+    }
+
     bool is_update() const {
         return f.name().find("update") != std::string::npos;
     }
@@ -1114,12 +1120,16 @@ class GPUTilingDedup {
                 continue;
             }
 
+            if (is_compute_at) continue;
+
             // Skip all gpu_blocks if the current Stage is "compute_at" another
             // stage, in which the gpu_blocks are already specified.
-            if (!is_compute_at && is_outer(v_name)) {
+            // if (is_outer(v_name) || is_parallel(v_name)) {
+            if (is_outer(v_name)) {
                 // Mark as gpu blocks;
                 f.gpu_blocks(v);
                 sched.push_schedule(f.name(), stage_num, "gpu_blocks(" + v_name + ")", {v_name});
+                continue;
             }
         }
     }
@@ -1193,6 +1203,7 @@ class GPUTilingDedup {
      */
     void has_split(const VarOrRVar &v, const VarOrRVar &vo, const VarOrRVar &vi, const Expr &factor, TailStrategy strategy) {
         debug(2) << f.name() << ".split(" << v.name() << "," << factor << ")\n";
+        is_split.emplace(v.name());
         outer_vars.emplace(vo.name());
         inner_vars.emplace(vi.name());
 
@@ -1247,7 +1258,7 @@ class GPUTilingDedup {
             sched.push_schedule(f.name(), stage_num, oss.str(), var_list);
         }
 
-        const bool is_already_split = (inner_vars.size() + outer_vars.size() > 0);
+        const bool is_already_split = (!is_split.empty());
         if (is_already_split) {
             // If the Mullapudi's auto-splitting algorithm already computes the
             // tile size, we simply mark the inner dims as gpu_threads();
@@ -1634,7 +1645,7 @@ struct Partitioner {
 
     // Loop over the dimensions of function stage 'f_handle' starting from innermost
     // and vectorize the first pure dimension encountered.
-    void vectorize_stage(
+    std::optional<pair<VarOrRVar, VarOrRVar>> vectorize_stage(
         const Group &g, Stage f_handle, int stage_num, Definition def,
         const Function &func, bool is_group_output, const Target &t, set<string> &rvars,
         map<string, Expr> &estimates, AutoSchedule &sched, GPUTilingDedup &gpu_tiling);
@@ -2824,11 +2835,11 @@ pair<VarOrRVar, VarOrRVar> Partitioner::split_dim(
     return make_pair(inner, outer);
 }
 
-void Partitioner::vectorize_stage(const Group &g, Stage f_handle, int stage_num,
-                                  Definition def, const Function &func, bool is_group_output,
-                                  const Target &t, set<string> &rvars,
-                                  map<string, Expr> &estimates, AutoSchedule &sched,
-                                  GPUTilingDedup &gpu_tiling) {
+std::optional<pair<VarOrRVar, VarOrRVar>> Partitioner::vectorize_stage(const Group &g, Stage f_handle, int stage_num,
+                                                                       Definition def, const Function &func, bool is_group_output,
+                                                                       const Target &t, set<string> &rvars,
+                                                                       map<string, Expr> &estimates, AutoSchedule &sched,
+                                                                       GPUTilingDedup &gpu_tiling) {
     vector<Dim> &dims = def.schedule().dims();
     int vec_dim_index = -1;
 
@@ -2902,7 +2913,11 @@ void Partitioner::vectorize_stage(const Group &g, Stage f_handle, int stage_num,
             debug(1) << "Outer dim vectorization of var \"" << vec_dim_name
                      << "\" in function \"" << f_handle.name() << "\"\n";
         }
+
+        return make_pair(inner, outer);
     }
+
+    return std::nullopt;
 }
 
 // Return true if the vars/rvars in 'ordering' are in the same order as the
@@ -3184,8 +3199,16 @@ void Partitioner::generate_group_cpu_schedule(
         }
     }
 
-    vectorize_stage(g, f_handle, g.output.stage_num, def, g_out, true, t,
-                    rvars, stg_estimates, sched, gpu_tiling);
+    {
+        auto vectorized_split = vectorize_stage(g, f_handle, g.output.stage_num, def, g_out, true, t,
+                                                rvars, stg_estimates, sched, gpu_tiling);
+
+        if (t.has_gpu_feature() && vectorized_split) {
+            auto [v_i, v_o] = *vectorized_split;
+            inner_dims.emplace_back(std::move(v_i));
+            outer_dims.emplace_back(std::move(v_o));
+        }
+    }
 
     // Parallelize definition
     Expr def_par = 1;
@@ -3296,14 +3319,15 @@ void Partitioner::generate_group_cpu_schedule(
             mem_handle = Func(mem.func).update(mem.stage_num - 1);
         } else {
             if (!outer_dims.empty()) {
+                string sanitized_g_out = get_sanitized_name(g_out.name());
                 if (tile_inner_var.is_rvar) {
                     Func(mem.func).compute_at(Func(g_out), tile_inner_var.rvar);
+                    debug(2) << mem_handle.name() << ".compute_at(" << sanitized_g_out << ", " << tile_inner_var.rvar << ")\n";
                 } else {
                     Func(mem.func).compute_at(Func(g_out), tile_inner_var.var);
+                    debug(2) << mem_handle.name() << ".compute_at(" << sanitized_g_out << ", " << tile_inner_var.var << ")\n";
                 }
 
-                string sanitized_g_out = get_sanitized_name(g_out.name());
-                debug(2) << mem_handle.name() << ".compute_at(" << sanitized_g_out << ")\n";
                 sched.push_schedule(mem_handle.name(), mem.stage_num,
                                     "compute_at(" + sanitized_g_out + ", " + tile_inner_var.name() + ")",
                                     {sanitized_g_out, tile_inner_var.name()});