diff --git a/Sources/SwiftFusion/Core/TypeKeyedArrayBuffers.swift b/Sources/SwiftFusion/Core/TypeKeyedArrayBuffers.swift
index 5d0a4a9a..a65be30f 100644
--- a/Sources/SwiftFusion/Core/TypeKeyedArrayBuffers.swift
+++ b/Sources/SwiftFusion/Core/TypeKeyedArrayBuffers.swift
@@ -87,6 +87,21 @@ extension TypeKeyedArrayBuffers {
     return _storage.allSatisfy { kv in  other._storage[kv.key]?.count == kv.value.count }
   }
 
+  /// Returns a mapping from each key `k` of `self` into `bufferTransform(self[k])`.
+  public func mapBuffers(
+    _ bufferTransform: (AnyArrayBuffer<ElementAPI>) throws -> AnyArrayBuffer<ElementAPI>
+  ) rethrows -> TypeKeyedArrayBuffers {
+    try .init(_storage: _storage.mapValues(bufferTransform))
+  }
+
+  /// Returns a mapping from each key `k` of `self` into the corresponding array, transformed by
+  /// `bufferTransform`.
+  public func compactMapBuffers(
+    _ bufferTransform: (AnyArrayBuffer<ElementAPI>) throws -> AnyArrayBuffer<ElementAPI>?
+  ) rethrows -> TypeKeyedArrayBuffers {
+    try .init(_storage: _storage.compactMapValues(bufferTransform))
+  }
+
   /// Returns a mapping from each key `k` of `self` into `bufferTransform(self[k])`.
   public func mapBuffers<NewElementAPI>(
     _ bufferTransform: (AnyArrayBuffer<ElementAPI>) throws -> AnyArrayBuffer<NewElementAPI>
@@ -102,6 +117,27 @@ extension TypeKeyedArrayBuffers {
     try .init(_storage: _storage.compactMapValues(bufferTransform))
   }
 
+  /// Returns the first key in `self` such that `predicate(self[k], other[k]) == true`, or `nil` if
+  /// no such key exists.
+  ///
+  /// - Requires: `self.hasSameStructure(as: parameter)`
+  public func firstBufferKey<OtherElementAPI, OtherConstruction>(
+    homomorphicArgument other: TypeKeyedArrayBuffers<OtherElementAPI, OtherConstruction>,
+    where predicate: (
+      _ myBuffer: AnyArrayBuffer<ElementAPI>,
+      _ otherBuffer: AnyArrayBuffer<OtherElementAPI>) throws -> Bool
+  ) rethrows -> TypeID? {
+    precondition(
+      _storage.count == other._storage.count,
+      "parameter must have same structure as `self`")
+    return try _storage.first { kv in
+      guard let v1 = other._storage[kv.key] else {
+        fatalError("parameter must have same structure as `self`")
+      }
+      return try predicate(kv.value, v1)
+    }.map { $0.key }
+  }
+
   /// Invokes `update` on each buffer of self, passing the buffer having the same `key` in
   /// `parameter` as a second argument.
   ///
diff --git a/Sources/SwiftFusion/Inference/TypeKeyedArrayBuffers+Vector.swift b/Sources/SwiftFusion/Inference/TypeKeyedArrayBuffers+Vector.swift
new file mode 100644
index 00000000..4528dd2a
--- /dev/null
+++ b/Sources/SwiftFusion/Inference/TypeKeyedArrayBuffers+Vector.swift
@@ -0,0 +1,57 @@
+// Copyright 2020 The SwiftFusion Authors. 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 _Differentiation
+import PenguinStructures
+
+extension TypeKeyedArrayBuffers: Equatable where ElementAPI: VectorArrayDispatch {
+  public static func == (lhs: Self, rhs: Self) -> Bool {
+    lhs.firstBufferKey(homomorphicArgument: rhs) { $0 != $1 } == nil
+  }
+}
+  
+extension TypeKeyedArrayBuffers: AdditiveArithmetic where ElementAPI == VectorArrayDispatch {
+  /// Returns the vector sum of `lhs` with `rhs`, where `lhs` and `rhs` are viewed as vectors in the
+  /// vector space direct sum of all the variables.
+  ///
+  /// Precondition: `lhs` and `rhs` have assignments for exactly the same sets of variables.
+  public static func + (lhs: Self, rhs: Self) -> Self {
+    lhs.updatedBuffers(homomorphicArgument: rhs) { $0 + $1 }
+  }
+
+  public static func += (lhs: inout Self, rhs: Self) {
+    lhs.updateBuffers(homomorphicArgument: rhs) { $0 += $1 }
+  }
+
+  public static func - (lhs: Self, rhs: Self) -> Self {
+    lhs.updatedBuffers(homomorphicArgument: rhs) { $0 - $1 }
+  }
+
+  public static func -= (lhs: inout Self, rhs: Self) {
+    lhs.updateBuffers(homomorphicArgument: rhs) { $0 -= $1 }
+  }
+
+  public static var zero: Self { .init() }
+}
+
+extension TypeKeyedArrayBuffers: Differentiable where ElementAPI: DifferentiableArrayDispatch {
+  public typealias TangentVector = MappedArrayBuffers<VectorArrayDispatch>
+  public mutating func move(along offset: TangentVector) {
+    updateBuffers(homomorphicArgument: offset) { $0.move(along: $1) }
+  }
+
+  public var zeroTangentVectorInitializer: () -> TangentVector {
+    { .init() }
+  }
+}