Two-Part Affine Approximation

• Two-part affine approximation is defined by decomposing a function into an affine component and a complementary nonlinear part to simplify analysis and control.
• In control-affine systems, the method enables the construction of predictors that remain affine in control inputs while capturing complex state dependencies.
• Techniques such as randomized feature mappings and intelligent breakpoint placement demonstrate improved efficiency and error control in high-dimensional and piecewise settings.

Two-part affine approximation designates approximation schemes that deliberately split a target into two interacting components and impose affine structure on one component, or else achieve affine approximation by a two-stage procedure. The clearest current formulation is in control-affine learning, where $\hat h(x,u)$ is required to be affine in $u$ while permitting arbitrary nonlinear dependence on the state $x$ (Kazemian et al., 2024). Closely related work applies the same split principle by decomposing a nonlinear multivariate function into unary or binary factors and replacing each factor by a piecewise affine approximation (Glunt et al., 2024), and by first establishing nearly affine behavior on one-dimensional slices before upgrading it to a genuinely high-dimensional affine approximation on a ball [(Li et al., 2012); (Hytönen et al., 2015)]. This suggests that the expression refers less to a single canonical object than to a recurrent approximation architecture.

1. Structural scope and recurrent patterns

Across the cited literature, two-part affine approximation appears in two main forms. In a structural split, the model itself is partitioned into an affine part and a non-affine part. In a procedural split, approximation is carried out in two stages, with local or partial affine control converted into a global approximation.

Setting	Two-part structure	Representative paper
Control-affine modeling	$f(x)$ plus $g(x)u$	(Kazemian et al., 2024)
Compositional PWA approximation	Decompose into unary/binary factors, then approximate each factor	(Glunt et al., 2024)
High-dimensional Lipschitz approximation	Control on 1D slices, then upgrade to a ballwise affine approximation	(Li et al., 2012, Hytönen et al., 2015)

In the control-affine case, the decomposition is explicit: $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$ Here $f(x)$ is the state-only term and $g(x)u$ is the control-affine term. In the compositional PWA case, the split is algorithmic: decompose a complicated nonlinear function into simpler unary or binary intermediates, approximate each intermediate function with a PWA map, then compose the approximants while propagating error bounds. In the high-dimensional Banach-space literature, the split is geometric: one first proves affine-like behavior on many one-dimensional restrictions and then uses multiscale aggregation to obtain a true affine approximation on a sub-ball.

This taxonomy does not define a single theorem. Rather, it isolates a recurring design principle: preserve or recover affine structure in one component while allowing complexity to reside in the other component or in the composition.

2. Control-affine random-feature models

The most literal use of two-part affine approximation appears in nonlinear modeling for control-affine systems (Kazemian et al., 2024). The modeling objective is to learn a function $\hat h:\mathcal X\times\mathcal U\to\mathbb R$ such that $\hat h(x,u)$ is affine in $u$0 while remaining nonlinear in $u$1. A concrete motivating example is the derivative of a certificate function: $u$2 which naturally decomposes into a state-dependent part and a control-affine part.

Two random-feature constructions are proposed. The Affine Dot Product (ADP) basis is

$u$3

and can also be written as a block-diagonal operator on the augmented control vector. Its dimension is $u$4. The Affine Dense (AD) basis is

$u$5

with output dimension $u$6. In both cases, the only dependence on $u$7 is through multiplication by $u$8, so the learned predictor is affine in the control input.

The state-dependent features are instantiated by random Fourier features: $u$9 with $x$0 sampled i.i.d. from $x$1. This yields Monte Carlo approximations of the associated state kernels.

The paper formalizes the representational content of these constructions through two compound kernels. The ADP basis approximates the ADP kernel

$x$2

while the AD basis approximates the AD kernel

$x$3

If each individual kernel approximation satisfies

$x$4

then both compound approximations satisfy

$x$5

The computational motivation is equally explicit. Kernel methods are described as typically requiring $x$6 time and $x$7 memory, whereas RF methods require about $x$8 time and $x$9 memory. Because the model remains affine in $f(x)$0, it can be inserted directly into a quadratic program in the certainty-equivalent setting, or an SOCP in the robust GP-style setting. On a double inverted pendulum, the nominal controller built from an incorrect model fails, while the data-driven affine models succeed in stabilizing or balancing the pendulum. ADP is described as more expressive but larger; AD is more compact, faster, and often competitive.

3. Compositional piecewise-affine approximation

A second major realization of the two-part pattern is the decomposition of a nonlinear multivariate function into one- and two-input factors, followed by separate PWA approximation of each factor and analytic propagation of the resulting error (Glunt et al., 2024). This avoids the combinatorial blow-up associated with direct high-dimensional PWA approximation.

A canonical example is

$f(x)$1

decomposed as

$f(x)$2

A higher-dimensional case study considers

$f(x)$3

which is decomposed into $f(x)$4 unary functions.

Two breakpoint-placement procedures are given for unary scalar-valued functions on an interval. Method 1 uses bisection to find the largest next breakpoint such that the secant-line error remains below a tolerance $f(x)$5. It can be nearly optimal in the number of breakpoints, but each interval error evaluation may require nonlinear optimization. Method 2 is optimization-free under the assumptions $f(x)$6 and a known bound

$f(x)$7

Its central bound is

$f(x)$8

The algebraic breakpoint rule updates $f(x)$9, allowing variable breakpoint spacing: when curvature is small, the next breakpoint can be farther away.

The second contribution of the paper is error propagation for compositions. For iterated compositions such as $g(x)u$0, the recursive structure is

$g(x)u$1

with $g(x)u$2 when the input is exact. The paper uses this to formulate two inverse problems: minimize complexity subject to a global error tolerance, or minimize error subject to a breakpoint budget.

The tower-function case study follows the workflow: propagate domains by interval arithmetic via CORA, sample tolerances logarithmically for each unary function, determine the number of breakpoints with Algorithm 1, form a mixed-integer optimization problem for the global complexity budget, and solve for a PWA approximation with a fixed number of breakpoints. The reported result is 163 breakpoints, upper error bound 0.4453, and true maximum error approximately 0.33; uniform breakpoint spacing with the same 163 breakpoints gives a worse true error, about 0.60. A common misconception is therefore that uniform gridding is an adequate proxy for affine segmentation. In the cited results, intelligent breakpoint placement is materially better.

4. Quantitative affine approximation in high-dimensional analysis

In Banach-space theory, the relevant question is not representation by finitely many affine pieces but the scale on which a Lipschitz map must resemble a single affine map. Li and Naor formalized this through the modulus of affine approximability $g(x)u$3, defined as the largest radius such that every Lipschitz $g(x)u$4 admits a sub-ball $g(x)u$5 and an affine map $g(x)u$6 with

$g(x)u$7

Their proof is explicitly interpretable as a two-part mechanism: first obtain one-dimensional control along many lines, then promote it to affine control on a genuinely high-dimensional cube or ball (Li et al., 2012).

The one-dimensional component is encoded by a dyadic coercive quantity

$g(x)u$8

and uniform convexity yields a rigidity inequality controlling deviation from the linear interpolant. The higher-dimensional component introduces multiscale quantities such as $g(x)u$9 and multilinearity defects $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$0; if $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$1 is small on a cube, then there exists an affine map $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$2 with

$$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$3

This is the geometric prototype of two-stage affine approximation: linewise straightness first, affine synthesis second.

For UMD targets, the scale is substantially improved (Hytönen et al., 2015). The main theorem states that there exists $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$4 such that for every $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$5, every $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$6-dimensional normed space $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$7, every $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$8-Lipschitz map $$\dot{x} = f(x) + g(x)u \qquad\text{or}\qquad x_{t+1} = f(x_t) + g(x_t)u_t.$$9, and every $f(x)$0, there exist an affine map $f(x)$1 and a sub-ball

$f(x)$2

such that

$f(x)$3

The proof passes through a Dorronsoro-type theorem for UMD targets and a canonical affine approximation operator

$f(x)$4

with

$f(x)$5

The analytic content is that a multiscale integral estimate for $f(x)$6 yields the existence of a scale and location at which the canonical affine part is already good.

These results do not assert that a high-dimensional Lipschitz map is globally close to affine. They provide explicit macroscopic sub-balls, but the radius remains exponentially small in the ambient dimension or in $f(x)$7.

5. Piecewise affine approximation in $f(x)$8 and $f(x)$9

A different but closely allied theory studies when rough functions can be approximated by countably piecewise affine maps. Kristensen and Rindler showed that $g(x)u$0 functions cannot, in general, be approximated well by piecewise constant functions, but can be approximated effectively by piecewise affine functions if the mesh is adapted to the singularities of the $g(x)u$1 function (Kristensen et al., 2012).

For a bounded Lipschitz domain $g(x)u$2, a finite Borel measure $g(x)u$3, and any $g(x)u$4, their main theorem states that for every $g(x)u$5 there exist a countable family $g(x)u$6 of rotated rectangles and simplices and a function $g(x)u$7 such that

• $g(x)u$8,
• $g(x)u$9 is affine for every $\hat h:\mathcal X\times\mathcal U\to\mathbb R$0,
• $\hat h:\mathcal X\times\mathcal U\to\mathbb R$1,
• $\hat h:\mathcal X\times\mathcal U\to\mathbb R$2, $\hat h:\mathcal X\times\mathcal U\to\mathbb R$3, and $\hat h:\mathcal X\times\mathcal U\to\mathbb R$4,
• $\hat h:\mathcal X\times\mathcal U\to\mathbb R$5.

The proof combines blow-up analysis at regular and singular points with an adapted local mesh construction. At $\hat h:\mathcal X\times\mathcal U\to\mathbb R$6-almost every point, the blow-up converges strictly in $\hat h:\mathcal X\times\mathcal U\to\mathbb R$7 to the affine tangent map. At $\hat h:\mathcal X\times\mathcal U\to\mathbb R$8-almost every point, the blow-up is one-directional and reflects Alberti’s rank-one structure. This is then matched by thin rectangles aligned with the singular direction, after which a gluing lemma constructs countably piecewise affine approximants on annuli with Whitney-type geometry.

For Sobolev functions, the same paper proves a genuine $\hat h:\mathcal X\times\mathcal U\to\mathbb R$9 estimate on a regular uniform triangulation $\hat h(x,u)$0: $\hat h(x,u)$1 where $\hat h(x,u)$2 is a $\hat h(x,u)$3-piecewise affine quasi-interpolant and $\hat h(x,u)$4 is an $\hat h(x,u)$5-modulus of continuity of $\hat h(x,u)$6. The authors emphasize that this is a full $\hat h(x,u)$7-error estimate rather than merely an $\hat h(x,u)$8-estimate. A frequent misunderstanding is that piecewise constants are a sufficient substitute in $\hat h(x,u)$9; the cited proposition shows that a nontrivial smooth compactly supported $u$00 cannot be area-strictly approximated by piecewise constant $u$01 functions.

6. Extensions, adjacent notions, and terminological boundaries

The same general preference for affine structure appears in several adjacent literatures. In two-stage adjustable robust optimization with covering constraints and right-hand-side uncertainty, an LP restriction produces an explicit feasible affine recourse rule

$u$02

and affine policies achieve an

$u$03

approximation guarantee (Housni et al., 2021). Here the “two-part” structure is first-stage static decisions plus second-stage affine recourse.

In propositional knowledge compilation, affine approximation means something different: the affine envelope $u$04 is the least affine Boolean function entailed by a Boolean knowledge base $u$05. The ROBDD-based algorithm of Zanuttini’s successors computes this envelope by translating the model set into a vector space over $u$06, taking XOR-closure, and translating back. On random functions, the reported average timings in milliseconds are $u$07 versus $u$08 at 12 variables and $u$09 versus $u$10 at 15 variables, while the model-set implementation ran out of memory beyond 15 variables (0804.0066).

A mathematically different two-part structure appears in the computation of Falconer’s affinity dimension for dominated affine iterated function systems. There the singular value function is split across exterior powers $u$11 and $u$12, and the resulting transfer-operator method yields super-exponentially accurate numerical approximations, including examples with more than 30 stable decimal digits (Morris, 2018). This is not a theory of affine approximation in the control or Banach-space sense, but it is an important adjacent use of two-part affine structure.

Another adjacent universality theorem proves that one arbitrary continuous non-affine univariate function together with the specific affine function

$u$13

generate, under addition and composition, a class dense in $u$14 for every compact $u$15, and similarly in $u$16 on $u$17 when the coordinate functions are available (Ismailov, 26 May 2026). This is again not the same formalism as control-affine or PWA approximation, but it shows how an affine generator can act as one half of a dense approximation mechanism.

Finally, the expression should not be conflated with the two-part MDL code of algorithmic information theory. The abstract of “Approximation of the Two-Part MDL Code” concerns successive monotonically length-decreasing two-part MDL codes, computability issues, and goodness of fit expressed through Kolmogorov complexity, not affine approximation [0612095]. The overlap is terminological rather than conceptual.

Taken together, these works show that two-part affine approximation is best understood as a family of approximation strategies centered on preserving affine structure where tractability, certification, or analytic control require it, while allowing the remaining component to carry the nonlinear or combinatorial complexity.