An Algorithm for Learning Switched Linear Dynamics from Data
Abstract
We present an algorithm for learning switched linear dynamical systems in discrete time from noisy observations of the system’s full state or output.
Switched linear systems use multiple linear dynamical modes to fit the data within some desired tolerance.
They arise quite naturally in applications to robotics and cyberphysical systems.
Learning switched systems from data is a NP-hard problem that is nearly identical to the k -linear regression problem of fitting k > 1 linear models to the data.
A direct mixed-integer linear programming (MILP) approach yields time complexity that is exponential in the number of data points.
In this paper, we modify the problem formulation to yield an algorithm that is linear in the size of the data while remaining exponential in the number of state variables and the desired number of modes.
To do so, we combine classic ideas from the ellipsoidal method for solving convex optimization problems, and well-known oracle separation results in non-smooth optimization.
We demonstrate our approach on a set of microbenchmarks and a few interesting real-world problems.
Our evaluation suggests that the benefits of this algorithm can be made practical even against highly optimized off-the-shelf MILP solvers.
Introduction
Switched linear systems model dynamical systems that arise in diverse areas including natural sciences (biological models) [4, 8, 15], robotics [2] and cyber-physical systems [1].
They are characterized by a finite set of modes, wherein each mode features a different set of governing equations for the future states in terms of the current state of the system.
In this paper, we study algorithms for inferring the dynamics of a switched linear system from data that consists of full-state observations (this can be subsequently extended to output observations using ARX models; see Section 2 for details).
Specifically, given a set of data points { ( x i , x ′ i ) } Ni =1 of current and next state observations, and a number of modes m , we wish to find matrices A 1 , . .
A m such that each data point ( x i , x ′ i ) is explained by some matrix A j : ( ∀ i ) ( ∃ j ) || x
′ i −
A j x i || ≤
ϵ || x i ||
+ τ , for given error tolerances ϵ, τ > 0 .
However, the switching signal which governs the current mode associated with each data point is latent.
This renders the problem NP-hard and the best known approaches have exponential complexity in terms of the number of data points and the desired number of modes [22].
The exponential complexity in the number of data points makes this problem especially challenging since many datasets of interest have thousands of data points but with tens of modes and state variables.
The key insight in this paper is that by reformulating this problem to incorporate a “gap” in the tolerance, we can significantly improve the complexity to be linear in the number of data points
N and exponential in the number of modes and the dimension of the state space.
Rather than providing a Y ES /N O answer for a single tolerance bound ϵ , our approach takes as inputs two levels of tolerance ϵ 1 < ϵ 2 . It either finds matrices that satisfy the “upper” tolerance bound ϵ 2 > 0 or returns N O if no set of m matrices can fit the data with the “lower” tolerance ϵ 1 ( < ϵ 2 ).
However, if the underlying data could have been modeled with some tolerance that lies in the gap ( ϵ 1 , ϵ 2 ) , our algorithm can either succeed or return N O with no guarantees provided.
Our technical approach exploits the gap in our formulation to argue that any set of solutions must have a lower bound on its volume.
Furthermore, by exploring the solution space in a carefully defined manner, we guarantee that each step of our approach will shrink the volume of the remaining solution space by a constant factor.
This is achieved by using ideas that are similar to separation oracles in combinatorial optimization [20], or the so-called “cutting plane” argument [10].
However, our approach applies these inside a tree-based branch-and-bound algorithm, wherein we use the volume contraction to prove bounds on the depth of the underlying tree.
Thus, we achieve a bound on the running time that is exponential in the state dimension and the number of modes but linear in the data size.
We demonstrate our ideas using a prototype implementation that is compared against a standard mixed-integer linear programming (MILP) formulation of the problem solved using a state-of-theart solver (Gurobi) [17].
Our experimental results confirm that the theoretical insights also apply in practice to yield an algorithm that is often orders of magnitude faster, especially as the amount of data increases.
We also demonstrate our approach using two interesting applications that include modeling human writing on a tablet in order to predict what alphabet is being written from a small number of samples, and deriving models of mechanical systems with contact forces from data.
1.1 Related Work
The problem of switched and hybrid system identification has been widely studied using a variety of approaches going back to the early 1980s; see, e.g., [32].
We discuss a few representative approaches, and refer the reader to the monograph by Lauer et al [22] for further details.
The identification approach may be exact (seeking a global solution that minimizes the error between the data and the model prediction), or approximate (wherein the optimization problem is solved approximately, or assumptions about the nature of the switching signal are used to simplify the problem).
Our approach here is exact but the problem itself is reformulated with a gap.
As for the nature of the switching mechanism, many approaches (including ours) focus on identifying the dynamics of individual modes while assuming that the switching signal is exogenous ; on the other hand, other approaches, such as piecewise affine system identification [5, 28], hybrid automata learning [29], or linear complementary systems learning [19], attempt to identify the dynamics of each mode along with rules for transitioning between modes.
Vidal et al [33] present an exact approach for finding matrices that fit the data in the absence of noise by posing the problem as one of finding zeros of a multivariate polynomial.
This is subsequently refined by Ozay et al [26] to handle the case of noisy data, by using sum-of-squares relaxations to obtain a semidefinite optimization problem.
Assumptions on the nature of the switching signal can drastically simplify the problem at hand.
For instance, Ozay [25] shows that the problem can be solved in polynomial time using a dynamic programming algorithm if the number of mode switches in a trajectory is bounded.
Other approximate approaches involve greedy algorithms [5] and block– coordinate descent (similar to k -mean regression) algorithms [21, 31].
However, the performance of these algorithms can vary depending on the nature of the data and no guarantees are available.
Machine learning techniques have also proved useful in hybrid system identification.
For instance, standard machine learning approaches (such neural networks) can be used to infer hybrid system models [16, 24].
However, the resulting number of modes can be exponential in the network size.
Moreover, switched systems often involve discontinuous switching which cannot be modeled adequately using standard activation functions such as sigmoids or ReLU.
Ly and Lipson present an approach for learning hybrid automata from data using evolutionary techniques [23].
Their approach combines symbolic regression for learning the dynamics of the individual modes with techniques for guessing the latent modes and inferring the conditions for switching from one mode to another.
Also relevant to this work, Dempster et al [12] study the problem of inferring hidden Markov models with linear systems, called Jump Markov Models (JMMs), using a modification of the Expectation–
Maximization (EM) algorithm. However, the E-step, which infers the assignments of modes, can be quite expensive for JMMs.
Subsequently, Gharamani and Hinton [14] propose a technique wherein the difficulties in the E-step are resolved using variational inference, whereas Blake et al [7] propose a sampling-based scheme.
A key difference between our approach and JMMs is that we focus on the algorithmic efficiency for learning the dynamics of the modes.
Although this requires us to infer the sequence of modes for the data, we do not learn the Markov model that generates this sequence.
However, our example on handwriting recognition presented in Section 4 illustrates how off-theshelf approaches for learning a finite-state model can be combined with our approach in a fruitful manner.
As for our application involving mechanical systems with contact forces (Section 4), a recent work by Jin et al [19] propose a convex relaxation approach to learn linear complementary systems .
These systems provide compact representations for a class of piecewise linear systems, including mechanical systems with contact forces [3, 27].
Our implementation does not use this compact representation, but is nevertheless able to learn switched linear models for these systems.
Finally, our approach side-steps the problem of estimating the number of modes which is often an issue.
This problem can be addressed by adding a penalty term accounting for the “model complexity” to the overall objective function to be minimized [23] or by using more sophisticated non-parametric Bayesian approaches wherein the prior distribution specifies unbounded number of modes and the varying number of model parameters varying as the number of modes increases [13].
Problem Formulation
We will use bold, lower case symbols ( x , y , w ) to denote vectors, and upper case letters ( A, B, C ) to denote matrices, unless otherwise mentioned.
Let R denote the set of real numbers, and N the set of natural numbers.
For m ∈ N , let [ m ] = { 1 , . . .
, m } .
Let E n denote the n × n matrix with all entries equal to 1 .
We write A ≤ B to denote that each entry of A is less than or equal to the corresponding entry of B (same notation will be used for vectors).
A linear dynamical system in discrete time is defined by its state x ( t ) ∈ R n and output y ( t ) ∈ R p , at each time t ∈ N .
The state evolves using the dynamical rule: x ( t +1) =
A x ( t ) , wherein A ∈ R n × n .
The output is obtained as y ( t ) =
C x ( t ) , wherein C ∈ R p × n .
Definition 1 (Switched Linear Dynamical System) .
A switched linear dynamical system with state x : N → R n and output y : N → R p with m ≥ 1 modes is defined by m matrices A 1 , . . .
, A m .
For each time t ∈ N , there exists a mode σ ( t ) ∈ [ m ] s.t.
x
( t + 1) =
A σ ( t ) x ( t ) .
The switching signal σ : N → [ m ] chooses a mode for each time t such that the continuous state x ( t ) evolves according to this mode.
The output is given by y ( t ) =
C x ( t ) according to a given matrix C ∈ R p × n .
Remark 1 .
Switched affine systems, with x ( t + 1) =
A j x ( t ) + b j for b j ∈ R n , are handled by augmenting x with an extra component that is always 1 , thus treating b j as a new column of A j .
The goal of this paper is to learn a switched linear dynamical system model from observation data.
The simplest type of data consists of full state observations involving pairs of states of the form { ( x i , x ′ i ) }
Ni =1 , wherein each x i =
x
( t i ) and x ′ i =
x ( t i + 1) are states observed at two successive time instants t i and t i + 1 .
In particular, the switching signal σ is not observed.
We assume that the number of modes m is known.
Remark 2 .
The assumption of full state observation is often impractical.
However, our techniques extend to output observations by using a fixed-length history of output observations
y ( t − p ) , . . .
, y ( t ) and finding a switched auto-regressive (SARX) model [35] of the form: y ( t + 1)
= (cid:88) p k
=0 B σ ( t ) ,k y ( t − k ) , wherein σ ( t ) ∈ [ m ] , B j,k ∈ R p × p .
Therefore, in the following, we will focus our description on the full-state data.
Definition 2 (Fitting With Error Bound τ and Tolerance ϵ ) .
A set of matrices A 1 , . . .
, A m “fits” the data D with error bound τ > 0 and tolerance ϵ > {+
+} iff
[Equation]
Our approach can be modified to handle L 2 or L 1 norm instead of L ∞ (see Appendix A).
The switched linear system identification (SLS-ID) problem inputs the number of modes m ≥ 2 , data D : { ( x i , x ′ i ) } N i =1 , error bound τ > 0 , tolerance ϵ > 0 and a magnitude bound γ .
It outputs a set of matrices A 1 , . . .
, A m that fits D with error bound τ and tolerance ϵ and satisfies
[Equation]
e, each entry of A j lies in [
[Equation]
or outputs I NFEASIBLE if no such matrices exist.
The SLS-ID problem is known to be NP-hard [22,
Theorem 5.1].
We formulate below a mixed-integer linear program (MILP) for the SLS-ID problem.
We choose mn 2 decision variables which form the unknown entries of the matrices
A 1 , . . . ,
A m .
We also use binary indicator variables w i, 1 , . . .
, w i,m ∈ { 0 , 1 } wherein i = 1 , . .
. , N ranges over each data item in D .
The indicator variable w i,j denotes that the i th data item in D is fitted by the matrix A j .
Each data item must be fitted by one matrix, i.e., (cid:80) mj
=1 w i,j = 1 , ∀ i ∈ [ N ] .
We add constraint (2) to set limits on the magnitude of the entries of the matrices A j .
Finally, we add the constraint that each data item is fitted by the corresponding matrix:
[Equation]
Here, M > 0 is a constant that is larger than γ ∥ x i [-k 
-]∥ 1 + ∥ x [-
-]′ i ∥ ∞ for all ( x i , x ′ i ) ∈ D , implying that (3) is trivially satisfied if w i,j = {+
 (so-called “big-+} M ” trick in integer linear programming).
The set of constraints above along with w i,j ∈ { 0 , 1 } form the constraints of a MILP.
The objective can be set in many ways: for instance, minimize the sum of the absolute value of the entries of each matrix, as a regularization term.
Lemma 1.
A set of matrices A 1 , . . .
, A m is feasible for the MILP defined above (for some valuation of the binary variables w i,j ) iff it fits the data D with error bound τ and tolerance ϵ .
The branch-and-bound algorithm for solving MILPs will solve m N linear programs in the worstcase, noting that for each i ∈ [ N ] exactly one binary variable in the set { w i, 1 , . . , w i,m } can be set to 1 .
This is exponential in the data size N ( = | D | ).
This bound is seemingly independent of the dimension n of the underlying state-space.
However, if N < mn , we are estimating more unknowns than the available data.
Therefore, in practice, it generally holds that N ≫ mn .
2.1 Identification with a Tolerance Gap
We now formulate a “relaxed version” of the SLS-ID problem with a “tolerance gap”, that will be called the SLS-ID-GAP problem.
Inputs: Number of modes m ≥ 2 , data D : { ( x i , x ′ i ) } Ni =1 , absolute error bound τ > 0 , two relative error tolerances ϵ 2 > ϵ 1 ≥ 0 and a magnitude bound γ .
Let ϵ gap = ϵ 2 − ϵ 1 .
For technical reasons that will be explained later, we assume that γ > ϵ gap .
Output: F EASIBLE along with a set of matrices A 1 , . . .
, A m satisfying the bounds constraints (2) that fits D with error bound τ and tolerance ϵ 2 , or, I NFEASIBLE if no set of m matrices with magnitude bound γ − ϵ gap can fit the data with tolerance ϵ 1 and error bound τ .
For any given data set D and tolerance ϵ 2 > 0 , we define ϵ min ( D, ϵ 2 ) ∈ [0 , ∞ ] as the least tolerance ϵ for which D can be fitted with fixed error bound τ and magnitude bound γ − ( ϵ 2 − ϵ ) .
Thus, the algorithm for the SLS-ID-GAP problem guarantees that (a) if ϵ min ( D, ϵ 2 ) ≤ ϵ 1 then the algorithm will return FEASIBLE with matrices that fit the data with tolerance ϵ 2 and magnitude bound γ ; (b) if ϵ min ( D, ϵ 2 ) > ϵ 2 , the algorithm will return INFEASIBLE ; (c) if ϵ min ( D, γ 2 ) ∈ ( ϵ 1 , ϵ 2 ] , the algorithm may return either answer without any guarantees.
The main result of the paper is as follows:
Theorem 1.
There exists an algorithm for solving the SLS-ID-GAP problem with complexity
O (cid:0) m Cmn 3 | log( nγ/ϵ gap ) | N poly ( m, n ) (cid:1) , wherein C is a constant factor and poly ( m, n ) is polynomial function of m and n .
The significance of this approach is that it is linear in the size of the data set N , although exponential in the number of modes m and the dimension of the state space n .
Tree-Based Exploration Algorithm
We now present, in stages, our algorithm for finding a set of matrices A 1 , . . .
, A m that fits the data.
First, we present the algorithm as a tree-based exploration approach, wherein each node of the tree Data: m, D, τ, ϵ 1 , ϵ 2 , γ (see Section 2.1). Result: Y ES with matrices A 1 , . . . , A m that fit the data with tolerance ϵ 2 , or N O .
1 Initialize tree T with a root node (see text for details) 2 while there exist unexplored leaf nodes in T do ν ← unexplored leaf node in T Mark ν as explored result ← Expand ν using Algorithm if result = ⟨ F EASIBLE , ( A 1 , . . . , A m ) ⟩ then return ⟨ Y ES , A 1 , . . . , A m ⟩ /* Solution discovered */ 8 return N O /
* No nodes remain to be explored */
Data: Leaf node ν with polyhedra P 1 , . .
, P m , data set U and assignment map µ .
Result:
New leaf nodes, or matrices Q 1 , . .
, Q m that fit all the data.
1 Choose feasible solutions (matrices) Q 1 , . . .
, Q m s.t.
for all j ∈ [ m ] , Q j ∈ P j (Section 3.2 will specify how to choose Q 1 , . .
, Q m ) 2 Find ( x i , x ′ i )
∈
U s.t.
for all j ∈ [ m ] , ∥ x ′ i − Q j x i ∥ ∞ > ϵ 2 ∥ x i ∥ ∞ + τ 3 if not found then return F EASIBLE , ( Q 1 , . .
, Q m ) /
* Q 1 , . . .
, Q m fit all the data */ 5 else ˆ U ← U \ { ( x i , x ′ i ) } /
* Remove ( x i , x ′ i ) from unassigned data */ for j ∈ [ m ] do /*
Constrain polyhedron P
j s.t. matrix
A j “fits” ( x i , x ′ i )
*/ ˆ P j ← P j ∩ { A j : ∥ x ′ i − A j x i ∥ ∞ ≤
ϵ 2 ∥ x i ∥ ∞ + τ } if ˆ P j contains a L ind ∞ -norm ball of radius ϵ gap (Lemma 3 explains why
then Create new child node ν j of ν .
Associate ν j with polyhedra P 1 , . .
, P j − 1 , ˆ P j , P j +1 , . .
, P m ˆ µ j ← µ ∪ { ( x i , x ′ i ) (cid:55)→ j } /* Associate ( x i , x ′ i ) with mode j .
/ Associate node ν j with set ˆ U and map ˆ µ j
represents a set of m convex polyhedra.
Subsequently, we will show how a careful choice of the tree exploration strategy will guarantee a bound on the maximum length of each branch of the tree.
This, in turn, yields the desired algorithm and its complexity guarantee.
The proofs of all the lemmas are outlined: detailed proofs are provided in Appendix B (cf. supplementary material).
The central data structure maintained by our algorithm is a tree T . Each node of the tree has the following associated information:
Convex polyhedra P 1 , . .
, P m , represented as systems of linear inequalities. Each P j ⊆ R n × n describes a set of possible solutions for matrix A j . 2.
A subset U ⊆ D containing data points that have not been “assigned” a mode yet.
A assignment map µ : ( D \ U ) →
[ m ] mapping each data point in D \ U to a mode in [ m ] .
Root Node: The root of the tree is initialized by m convex polyhedra P 1 , . .
, P m , wherein P j encodes the bounds on each entry of the matrix
A j : − γE n ≤ A j ≤ γE n .
The set U at the root is the full data set D and the associated map µ is the “empty map” since its domain is empty.
3.1 Expanding a Node
Starting from the root node, our algorithm iteratively expands the tree T until no unexplored leaf remains or a suitable set of matrices is found (see Algorithm 1).
Expansion : Each expansion step involves choosing an unexplored leaf ν of the tree and carrying out the steps outlined in Algorithm 2.
Namely, expanding ν begins with choosing feasible solutions
Q 1 , . . , Q m from polyhedra P 1 , . . .
, P m , respectively (line 1).
The choice of these solutions will be described in detail in Section 3.2. Next, we scan through all the “unassigned” data U to find a data point ( x i , x ′ i ) ∈ U that cannot be fitted by Q 1 , . .
, Q m (line 2).
If no such data point can be found then we have, in fact, fitted all the data using Q 1 , . . .
, Q m .
Therefore, we can terminate (line 4).
If a data point ( x i , x ′ i ) [-
-]∈ U is found in the previous step, then we create m potential child nodes ν 1 , . .
, ν m for the current node.
Each potential child ν j will have associated polyhedra P 1 , . .
, P j − 1 , ˆ P j , P j +1 , . .
, P m , wherein, for j ′ ̸ = j , P j ′ remains the same as in node ν , and ˆ P j is defined in order to force the matrix A j to fit the data point ( x i , x ′ i ) that could not be fitted by the previous candidate matrices (line 8).
Node ν j will have associated data set ˆ U : U \ { ( x i , x ′ i ) } and mode assignment map ˆ µ j : µ ∪ { ( x i , x ′ i )
(cid:55)→ j } . Finally, the potential child node ν j is actually added to the tree only if ˆ P j satisfies some lower bound on its size (line 9).
Thus, the process of expanding a node of the tree results either in termination with a feasible solution
Q 1 , . .
, Q m that fits the entire data, or in the possible addition of at most m new leaf nodes to the tree.
We will now establish important properties of this algorithm.
Lemma 2.
Let Q 1 , . . .
, Q m be the chosen candidates while expanding some node ν .
For each child ν j of ν with associated polyhedra P 1 , . . .
, P j − 1 , ˆ P j , P j +1 , . .
, P m , we have that Q j ̸∈ ˆ P j .
The proof (see Appendix B) shows that the constraints added to (cid:99) P j in line 8 are not satisfied by Q j . Recall that the induced (operator) L ∞ norm of a matrix: ∥ A ∥ ind ∞ . = max ∥ x ∥ ∞ ≤ 1 ∥
A x ∥ ∞ .
Given
A ∈ R n × n and ϵ ≥ 0 , let B ind ∞ ( A, ϵ ) denote the ball of radius ϵ (w.r.t. ∥·∥ ind ∞ ) centered at A . We prove that the tree-based exploration algorithm is complete for the SLS-ID-GAP problem: if there is a set of matrices A ∗ 1 , . . . , A ∗ m fitting the data with tolerance ϵ 1 and satisfying the reduced magnitude bound γ − ϵ gap , then there is an unexplored leaf whose associated polyhedra contain these matrices.
Lemma 3.
At each iteration of the algorithm, there exists an unexplored leaf ν in the tree with associated polyhedra P 1 , . .
, P m such that for all j ∈ [ m ] , B ind ∞ ( A ∗ j , ϵ gap ) ⊆ P j .
The proof (see Appendix B) is by induction on the number of iterations of the algorithm.
At the beginning, we show that the root node satisfies the property of the lemma.
Then, we prove that, at each iteration, if a leaf satisfying the property is picked to be expanded, then at least one child of the leaf is created and “inherits” the property.
The requirement that γ > ϵ gap is key for this lemma.
From the definition of the child nodes (line 9), the convex sets associated to each leaf of the tree satisfy a lower bound on their volume.
Lemma 4.
At each iteration of the algorithm, and for each leaf ν in the tree, with associated polyhedra P 1 , . . .
, P m , it holds that for all j ∈ [ m ] , vol ( P j ) ≥ (2 ϵ gap ) n 2 ( n !) n .
[-The crux of the proof is to show that if a matrix A j satisﬁes k A j  A ⇤ j k ind 1  ✏ gap , then we can showthat, for each data item ( x i , x 0 i ) 2 D with µ ⇤ ( x i , x 0 i ) = j , k x 0 i  A j x i k 1  k x 
 i  A-]Proof.
Each P j contains a L ind ∞ -norm ball of radius ϵ gap[-k x i k 
  ✏ 2 k x k 1 + ⌧-] .
Such a ball is the product of n unit L 1norm balls (one for each row) each scaled with a factor ϵ gap .
The rest follows from the well-known formula for the volume ( 2 n n ! ) of a unit L 1 -norm ball in n dimensions.
We have not proven a bound on the size of the tree explored by our algorithm so far.
In the next subsection, we describe a way of selecting the candidates Q 1 , . . . Q m (line 1) such that the volume of ˆ P j will be smaller than some fraction α < 1 times the volume of P j .
This, combined with the lower bound on the volume of these sets (Lemma 4), will provide a bound on the length of each branch (i.e., the depth) of the tree.
We prove an effective bound on the depth of the tree using the so-called cutting-plane argument from non-smooth optimization [10].
First, we refine line 1 of our algorithm wherein we choose candidates Q j ∈ P j for each j ∈ [ m ] .
Specifically, we will choose Q j as the center of the maximum volume inscribed ellipsoid (MVE) of P j .
The MVE center of a polyhedron can be computed efficiently by using semi-definite programming (SDP) [6, Proposition 4.9.1].
Now consider the child node ν j of ν , to which we associate the polyhedron ˆ P j ⊊ P j .
We show that the volume reduces by at least a factor α . = (1 − 1 n 2 ) < 1 .
Lemma 5.
vol ( ˆ P j ) ≤ (1 − 1 n 2 ) vol ( P j ) . The above follows from the classical cutting-plane argument [30] applied on the polyhedron ˆ P j , which is obtained by cutting P j with a hyperplane excluding the MVE center (see Appendix B).
Lemma 6.
The depth of the tree is O ( mn 4 log( nγ/ϵ gap )) .
The proof (see Appendix B) observes that each time we expand a node in the tree, one polyhedron has its volume reduced by a factor α as stated in Lemma 5.
Also, when the overall volume becomes smaller than V min as in Lemma 4, the node is pruned.
This places a bound on the depth of the tree, as stated in the lemma.
In fact, the n 4 term is reduced to n 3 using the observation that each polyhedron
P j for each node ν is the Cartesian product of n polyhedra P j,k , for k ∈ [ n ] , involving the variables from the k th row of the unknown matrix A j .
This “fine-grained” analysis is provided in Appendix C. Theorem 2.
The overall size of the tree cannot exceed m O ( mn 3 log( nγ/ϵ gap )) nodes, wherein the complexity of expanding each node is linear in the size N of the data set and involves solving m
SDPs each with n 2 variables and O ( mn 3 ) constraints.
Experimental Evaluation
In this section, we will describe an evaluation of our approach meant to answer two key questions: (a) Do the theoretical guarantees translate into superior empirical performance when compared to highly optimized MILP solvers? (Namely, we compare our approach against the MILP solver
Gurobi [17] over a set of “microbenchmarks” of varying dimensions, number of modes and data sizes.); (b) Does the approach yield interesting results on real-life datasets?
(Namely, we illustrate our approach on datasets from handwritten alphabets and mechanical systems with contact forces.)
Implementation: We implemented the proposed approach in Python 3.8, using Gurobi [17] to encode and solve linear programs.
We assume that the number of modes m (or an upper bound on it) is given.
Also, our implementation fixes ϵ 1 = 0 and ϵ 2 = ϵ gap = ϵ > 0 .
Fixing ϵ 1 = 0 implies that our algorithm either finds matrices that fit the data with tolerance ϵ 2 or concludes that no matrices exist that fit the data with zero relative error tolerance.
Therefore, throughout this section, we will report the value of ϵ .
Our implementation coincides with the description in Section 3 with one important modification: we compute the Chebyshev center (center of the largest inscribed ball ) instead of the center of the maximum volume inscribed ellipsoid (MVE).
The Chebyshev center can be computed very efficiently and reliably using Linear Programming [9], and provides a good approximation of the MVE center.
Although the theoretical guarantees on the termination of the process using the Chebyshev center are weaker than those with the MVE center (Lemma 5), the use of Chebyshev center in this context is a widely-used heuristic [10, §4.4]. The containment check described in line 9 of Algorithm 2 is implemented by comparing the Chebyshev radius against ϵ gap .
Evaluation on Microbenchmarks: We compare the proposed approach against the MILP approach on a suite of synthetic microbenchmarks.
Each such benchmark consists of m randomly chosen n × n Hurwitz matrices, where m, n are varied systematically.
A total of N data points were generated for each experiment from k = N/T trajectories, each of length T = 10 time steps starting from a random initial state in [ − 1 , 1] n .
An additive noise sampled uniformly at random in the range [ − 0 . 05 , 0 . 05] was added to each state.
In Figure 1, we illustrate how the two approaches scale in terms of computation time with respect to the number of data points ( N ), number of states ( n ), and number of modes ( m ). All timings are averaged over 10 separate runs to account for the variability in computation times. Further details on microbenchmark generation are provided in Appendix D.
The comparison clearly shows that the MILP solver’s computation time increases rapidly with the number of data points N . In contrast, the running time of our approach depends linearly on N .
Both approaches scale similarly in terms of number of modes ( m ) and dimension ( n ).
[Figure]
[Figure] [Figure]
[Table]
We compare the accuracy of the proposed approach against k -Linear Regression ( k LR) [21] in terms of one-step prediction errors: min mj =1 || x ′ i − A j x i || ∞ , measured over a held out test dataset.
Tablecompares the prediction errors for a set of 3 microbenchmarks. The proposed approach has smaller error bounds compared to k LR but takes more time, as expected.
Handwriting Recognition: We now evaluate our approach on a dataset from human handwriting on a tablet: our goal is to identify various modes with dynamics that describe how letters are traced out on the tablet.
We generate our own handwriting dataset by having an author trace out the letters “a”, “b”, “c” and “d” using their fingers on the mouse pad of their laptop.
We collect the ( x, y ) locations of the handwritten letters over time.
We generate our final dataset by interpolation in order to ensure that the samples are roughly equidistant from each other. This can be done in an online fashion, if need be.
It ensures that our model here is not capturing artifacts of the pressure sensor in the mouse pad or how the person tracing the alphabet may speed-up or slow-down at various points in time.
We also scale the ( x, y ) values so that the letters fit within a square of side length 1 .
Further details are provided in Appendix F. We collect 10 samples for each letter (Figure 2-left).
We apply our approach to learn m = 3 matrices that fit the data with ϵ = 0 . 05 and τ = 0 .
For each sample, we obtain a corresponding switching sequence from this process.
We apply a wellknown grammatical inference approach to learn a Jump Markov Model (JMM), wherein the various dynamical modes are represented on the edges of the automaton [11, 34].
The estimated mode for each data point of each letter is represented in Figure 2-left.
Each color represents a different mode.
We note that the modes naturally corresponds to different parts of the letters.
Figure 2-middle shows an example of JMM for the letter “b”.
Each transition in the JMM represents a mode and a probability of taking the edge.
At state q 0 , mode 1 is observed for 92 % of the time, and at q 1 , mode 0 is observed for 93 % of the time.
We then try to predict a class for an alphabet as it is being written, based on the estimated JMMs (see Figure 2-right).
Each column represents 20 % , 40 % , 60 % , 80 % , 95 % portion of a letter.
We use the JMMs and Bayesian inference to compute the probability of a letter given a partial sequence of states.
The letters “a”, “b” and “d” are correctly predicted but the letter “c” is classified as an “a”: a visual inspection of Figure 2-right explains why this is the case.
In conclusion, we show that our switched linear system identification algorithm works on the noisy handwriting data and can be used to identify a JMM from sequences of identified modes which in turn can be useful for data classification and prediction.
Acrobot: We evaluate our approach on an acrobot benchmark with soft joint limits [3].
We sampled N = 30 trajectories with time step dt = 0 . 03 seconds for T = 3 seconds.
We then identified the dynamics at each mode using the proposed approach (with m = 3 , τ = 0 . 01 , and ϵ = 0 . The average one-step prediction error 5 of the proposed approach on a held-out test dataset consisting of 5 trajectories is 0 .
005 (min: 0 . 0 , max: 0 . 18 ).
The average training time is 7 . 56 seconds. In comparison, the average one-step prediction error of k -Linear Regression( k LR) [21] on the same dataset is 1 .
80 (min: 0 . 0 , max: 8 . 68 ). The average training time of k LR is 0 .
18 seconds. Figure 4 shows the predictions of the proposed approach on a sample test trajectory. Despite the underlying system having an infinite number of modes (as the time-sampled system of a continuous-time hybrid linear system), our system identification technique is able to identify three main linear modes that explain most of the data both in the training and test datasets. Cartpole: We evaluate our approach on a cartpole system with soft walls [3].
We sampled N =trajectories with time step dt = 0 . 05 seconds for T = 5 seconds. We then identified the dynamics at each mode using the proposed approach (with m = 3 , τ = 0 . 1 , and ϵ = 0 . 1 ).
The average one-step 5 One-step prediction error: min mj = || x − A x i || ∞ prediction error of the proposed approach on a held-out test dataset consisting of 5 trajectories is 0 . 016 (min: 0 . 00 , max: 0 . 15 ).
The average training time is 9 .
35 s. The average prediction error of k -Linear Regression( k LR) [21] is 0 . 04 (min: 0 . 0 , max: 0 . 15 ) and the average time taken by k LR is 0 . 16 seconds. Additional plots are provided in Appendix E. Similarly to the acrobot example, the learned system is able to explain most of the data (both in the training and test datasets) with a few (three) linear modes, despite the fact that the actual system has an infinite number of modes (as the time-sampled system of a continuous-time hybrid linear system).
Conclusions: Limitations, Future Work and Broader Impacts
In summary, we have presented an approach that improves the time complexity of switched linear system identification using a novel problem formulation with a “gap” and using ideas from the ellipsoidal method in combinatorial optimization.
The resulting complexity is linear in the number of data points.
Limitations include (a) Our approach uses full state information and extends to output observations using SARX models.
However, the number of modes in the SARX model can be exponentially larger when compared to the state-space model.
(b) Our approach finds the matrices fitting dynamics but does not infer the “generator” for the mode switches, although we present an example using Jump Markov Models.
However, the problem of learning the matrices for various modes in conjunction with the “generator” may not always lend itself to a clean separation wherein the modes are first inferred using our approach and a generative model such as a Markov chain is then inferred.
For instance, our approach may find matrices that fit the data well but lead to poorly fitting system models.
We will study the use of backtracking in our approach to generate multiple “dissimilar” solutions; (c) Finally, our implementation performs the tree exploration without leveraging other information about the system or using heuristics that have proved powerful in branch-and-bound algorithms.
Nevertheless, it provides a proof of concept that it can be competitive against highly-optimized MILP solvers.
We will expand on these ideas in our future work.
In terms of broader impacts, model identification is a very important machine learning problem that has very important and beneficial applications in areas such as controls, autonomous systems and medical applications. Better modeling of physiological processes can lead to improved closedloop medical devices.
Improved ability to predict human movements can lead to safer human-robot interactions.
However, our approach can be used to improve surveillance systems that could be themselves be used in ways that are detrimental to the freedom of individuals and societies.
Acknowledgments: We thank the anonymous reviewers for their detailed comments and suggestions.
This research was funded in part by the Belgian-American Education Foundation (BAEF) and the US National Science Foundatton (NSF) under award numbers 1836900 and 1932189.
A Problem Formulation using L 1 and L 2 norms
Our approach essentially amounts to computing convex sets P j ⊆ R n × n containing the feasible values for the matrices A j .
If the L ∞ or L 1 norm is used for the constraint (1) on A j , then the sets P j are polyhedra. In fact, if the L ∞ norm is used, the set P j can even be described as the Cartesian product of n polyhedra in R n (one for each row of A j ). If the L 2 norm is used in (1), then the convex sets can be described using second-order cones and linear constraints.
In any case, that is, for any vector norm ∥·∥ , the convex set P j defined by the constraints (1) with the appropriate norm satisfies that B ind
( A ∗ j , ϵ gap ) ⊆
P j , where ϵ gap =
ϵ − ϵ 1 and A ∗ j explains the data with error bound τ and tolerance ϵ 1 .
Here, B ind ( A ∗ j , ϵ gap ) is the ball in R n × n centered at A ∗ j with radius ϵ gap w.r.t.
the matrix norm ∥·∥ ind induced by ∥·∥ (the proof is identical to the one of
Lemma 3).
Lemma 4 then applies mutatis mutandis using the volume of B ind (0 , ϵ gap ) .
Finally, regarding the computation of the MVE centers of the sets P j (which is a key step of Algorithm 2, as it is used to compute the candidate matrices A j in line 1): finding the MVE center of convex sets described by linear and second-order cone constraints can be cast as a semidefinite optimization problem [9, § 8.2.4], so that it can be solved efficiently.
B Missing Proofs From Section
Proof of Lemma 2.
Let U be the data set associated to ν .
Since we did not terminate, there is ( x i , x ′ i ) ∈
U such that ∥ x ′ i − Q j x i ∥ ∞ > ϵ 2 ∥ x i ∥ ∞ + τ .
However, ˆ P j is obtained by adding the constraint ∥ x ′ i − A j x i ∥ ∞ ≤ ϵ 2 ∥ x i ∥ ∞ + τ to the existing constraints in P j .
Therefore, Q j ̸∈ ˆ P j since it violates the new constraint.
Proof of Lemma 3.
First, we prove that the property holds for the root node.
Therefore, let A j ∈ B ind ∞ ( A ∗ j , ϵ gap ) . By the formula for the induced norm, it holds that for any matrix M , if ∥ M ∥ ind ∞ ≤ ϵ , then for all k 1 , k 2 ∈ [ n ] , | M k 1 ,k 2 | ≤ ∥ M k 1 : ∥ 1 ≤ ϵ , so that − ϵE n ≤ M ≤ ϵE n . Hence, letting
M .
= A j − A ∗ j and ϵ . = ϵ gap , we get that − ϵ gap E n ≤ A j − A ∗ j ≤ ϵ gap E n .
Thus, − γE n ≤ A j ≤ γE n , so that A j ∈ P j at the root node.
Suppose that at the beginning of the k th iteration, the unexplored leaf ν , with associated polyhedra P 1 , . . , P m , has the property that B ind ∞ ( A ∗ j , ϵ gap ) ⊆ P j for all j ∈ [ m ] .
We wish to prove the property for some unexplored leaf after the iteration. This is trivial if the leaf ν is not expanded in that iteration. Suppose the leaf ν is expanded. Let ( x i , x ′ i ) be the data point that cannot be explained by the candidates that were chosen, and let j ∈ [ m ] be such that A ∗ j explains the data point ( x i , x ′ i ) with tolerance ϵ 1 .
We show that B ind ∞ ( A ∗ j , ϵ gap ) ⊆ ˆ P j where ˆ P j is defined as in line 8. Indeed, let A j ∈ B ind ∞ ( A ∗ j , ϵ gap ) .
{+It holds that ∥ x ′ i − A j x i ∥ ∞ ≤ ∥ x ′ i − A ∗ j x i ∥ ∞ + ∥ ( A j − A ∗ j ) x i ∥ ∞ ≤ ∥ x ′ i − A ∗ j x i ∥ ∞ + ϵ gap ∥ x i ∥ ∞ ≤ ϵ 
+}
∥
x
i ∥ ∞ + τ + ϵ gap ∥ x i ∥ ∞ ≤ ϵ 2 ∥ x i ∥ ∞ + τ , where the second inequality comes from the definition of the induced norm and the assumption on A j , and the third inequality comes from the assumption on A ∗ j explaining ( x i , x ′ i ) with tolerance ϵ 1 .
Thus, B ind ∞ ( A ∗ j , ϵ gap ) ⊆ ˆ P j . Since, B ind ∞ ( A ∗ j , ϵ gap ) ⊆ P j , this implies that B ind ∞ ( A ∗ j , ϵ gap ) ⊆
ˆ P j , concluding the proof.
Proof of Lemma 5.
From Lemma 2, we note that Q j / ∈ ˆ P j .
In other words, ˆ P j ⊊ P j excludes the MVE center of P j . Following [30] (or [10, § 4.3] for a more recent reference), we have vol ( ˆ P j ) ≤ (1 − 1 d ) vol ( P j ) , where d is the dimension of P j . Here, d = n 2 , concluding the proof.
Proof of Lemma 6. Consider any path from the root to a leaf whose length is mK for some integer K > 0 . We note that for each node ν
and any of its children ν j , the polyhedron ˆ P j satisfies the inequality vol ( ˆ P j ) ≤ α vol ( P j ) , where α = 1 − 1 n 2 (Lemma 5).
Let us say that the index j ∈ [ m ] is refined by such an edge. By the pigeon-hole principle, for a path of length mK , there exists at least one index j that is refined K or more times along the path. Therefore, we have that: vol ( P ( K ) j ) ≤ α K vol ( P (0) j ) , where P (0) j is the j th polyhedron at the root and P ( K ) j is the j th polyhedron at the leaf.
We know that vol ( P (0) j ) = (2 γ ) n 2 . Thus, there exists K min such that for any K ≥ K min , vol ( P ( K ) j ) < V min and thus the branch will end up being “pruned” by our algorithm (line 9).
It holds that [Equation] where the last inequality follows from log(1 − 1 n 2 ) ≤ − 1 n 2 . Therefore, the depth is upper bounded by mK min =
mn 4 log( nγ/ϵ gap ) . C Fine-Grained Complexity Analysis of Tree Search As we noticed in Appendix A, for the L ∞ norm, each set P j can be described as the Cartesian product of n polyhedra in R 1 × n (one for each row of the matrix). The MVE center of a Cartesian product of convex sets is the vector containing the MVE center of each convex set.
Therefore, the volume reduction guarantee in Lemma 5 can be refined as: vol ( ˆ P j ) ≤ (1 − 1 n ) vol ( P j ) (see Lemma 8 below). By applying the same argument as in the proof of Lemma 6, we then get the bound O ( mn 3 log( nγ/ϵ gap )) on the depth of the tree. We will now present this in more details.
Let ν be any node of the tree and P 1 , . . , P m be the associated polyhedra. Lemma 7. Each P j can be written as a Cartesian product P j = P j, 1 × · · · × P j,n wherein each polyhedron P
j,i ⊆ R 1 × n involves just those decision variables of the matrix A j associated with its i th row.
Proof.
Proof is by induction. To begin with, we note that this is true for the root node of the tree. For the induction step, assume that the property is satisfied at some node ν of the tree and consider any of its child nodes ν j .
We note that, for any ( x i , x ′ i ) , the constraint ∥ x ′ i − A j x i ∥ ∞ ≤ ϵ 2 ∥ x i ∥ ∞ + τ is of the form || z || ∞ ≤ a for a vector z and scalar a .
This can be decomposed into constraints − a ≤ z i ≤ a for each row of z . Hence, the constraint ∥ x ′ i − A j x i ∥ ∞ ≤ ϵ 2 ∥ x i ∥ ∞ + τ can be decomposed into a conjunction of n constraints, each involving a different row of A j . From the induction hypothesis and the definition of ˆ P j (line 8), it follows that ˆ P j can be written as a Cartesian product of n polyhedra, each involving a different row of A j , concluding the proof. We now state a refined version of Lemma 5.
Lemma 8. vol ( ˆ P j ) ≤ (1 − 1 n ) vol ( P j ) . Proof. Let ˆ P j,i be the polyhedron associated with the i th row of matrix A j in the j th child node of some node ν such that the Q j,i / ∈
ˆ P j,i , wherein Q j,i denotes the i th row of candidate matrix Q j explored during the expansion of node ν by Algorithm 2. Since Q j,i is the MVE center of P j,i , it holds, by the cutting-plane argument (cf. proof of Lemma 5), that vol ( ˆ P j,i ) ≤ (1 − 1 n ) vol ( P j,i ) . Now, since vol ( ˆ P j ) =
(cid:81) ni =1 vol ( ˆ P j,i ) and vol ( P j ) = (cid:81) ni =1 vol ( P j,i ) , we get the desired result. Lemma 9. The depth of the tree is O ( mn 3 log( nγ/ϵ gap )) .
Proof. The proof is very similar to that of Lemma 6. The only thing that to be changed is the value of α = 1 − 1 n (instead of 1 − 1 n 2 ). We can then use the bound: log(1 − 1 n ) ≤ − 1 n , to get the desired result.
For Figure 1, we generate 90 microbenchmarks with varying values of n and m . We fixed the dynamics at each mode of the microbenchmark by sampling a random n × n Hurwitz matrix. The
Hurwitz matrices were generated by first generating random diagonal and invertible matrices of appropriate dimensions and then applying a similarity transformation on them.
We then generated trajectories from the microbenchmark by starting at some initial state in [ − 1 , 1] n and simulating the forward in time for T=10 time steps by randomly picking the mode at each time step. We added a uniform noise with amplitude ∈ [ − 0 . to all the trajectories. Figure 5 shows one such microbenchmark with n = 4 and m = 3 and some sample trajectories from the microbenchmark.
F Details on the Handwriting Recognition We generated a human handwriting dataset where each letter is drawn on a canvas of size 300 ×pixels. Therefore, we asked the users to write letters a, b, c, d and collected the locations ( x, y ) of the handwritten letters.
We made sure that each handwritten letter had enough raw points ( T raw > 100 ). Subsequently we interpolated the raw data points so that we could extract T = 30 data points that are roughly equidistant from each other. The complete interpolation and extraction process in explained below; see also Fig. 8 for an illustration.
Given the raw data points P : { p 1 , ..., p T raw } , with p i : ( x i , y i ) ∈ R 2 ,
Measure the total distance D =
(cid:80) i
{+d i where d i = (cid:112) ( x i +
+}
{+− x i ) 
+}
+ ( y i +1 − y i )2.
For each j ∈ { 0 , . . .
, T − 1 } , let d ′ j =
DT − 1 j , .
Compute the j th data point (ˆ x j , ˆ y j ) , as the point on the line that is at a distance d ′ j from p 1 .
Therefore:
(a) Find i ∈ { 1 , . .
, T raw − 1 } such that
(ˆ x j , ˆ y j ) is located between p i and p i +1 ,
i
e., such that the following condition holds true,
[Equation]
(b) Define (ˆ x j , ˆ y j ) : (1 − t ) p i
+ tp
{+i +1 , where t = ( d ′ j − (cid:80) ii =
+}
d i ) /d
{+i . Since (ˆ x j , ˆ y j ) = p i + t ( p i +
+}
− p i ) , this satisfies that requirement that (ˆ x j , ˆ y j ) is at a distance d ′ j from p 1 . We collected 10 samples for each letter and each letter’s final data size was set to T = 30 . As a result, the total number of data points is N = kT = 10 · 30 = 300 .