# Efficient $\ell_{1}/\ell_{q}$ Norm Regularization

Jun Liu  Jieping Ye

(September 23, 2010)

###### Abstract

Sparse learning has recently received increasing attention in many areas
including machine learning, statistics, and applied mathematics. The mixed-
norm regularization based on the $\ell_{1}/\ell_{q}$ norm with $q>1$ is
attractive in many applications of regression and classification in that it
facilitates group sparsity in the model. The resulting optimization problem
is, however, challenging to solve due to the structure of the
$\ell_{1}/\ell_{q}$-regularization. Existing work deals with special cases
including $q=2,\infty$, and they can not be easily extended to the general
case. In this paper, we propose an efficient algorithm based on the
accelerated gradient method for solving the $\ell_{1}/\ell_{q}$-regularized
problem, which is applicable for all values of $q$ larger than $1$, thus
significantly extending existing work. One key building block of the proposed
algorithm is the $\ell_{1}/\ell_{q}$-regularized Euclidean projection (EP1q).
Our theoretical analysis reveals the key properties of EP1q and illustrates
why EP1q for the general $q$ is significantly more challenging to solve than
the special cases. Based on our theoretical analysis, we develop an efficient
algorithm for EP1q by solving two zero finding problems. Experimental results
demonstrate the efficiency of the proposed algorithm.

##  1 Introduction

Regularization has played a central role in many machine learning algorithms.
The $\ell_{1}$-regularization has recently received increasing attention, due
to its sparsity-inducing property, convenient convexity, strong theoretical
guarantees, and great empirical success in various applications. A well-known
application of the $\ell_{1}$-regularization is the Lasso [32]. Recent studies
in areas such as machine learning, statistics, and applied mathematics have
witnessed growing interests in extending the $\ell_{1}$-regularization to the
$\ell_{1}/\ell_{q}$-regularization [2, 7, 14, 23, 29, 37, 38]. This leads to
the following $\ell_{1}/\ell_{q}$-regularized minimization problem:

$\min_{\mathbf{W}\in\mathbb{R}^{p}}f(\mathbf{W})\equiv
l(\mathbf{W})+\lambda\varpi(\mathbf{W}),$ (1)

where $\mathbf{W}\in\mathbb{R}^{p}$ denotes the model parameters, $l(\cdot)$
is a convex loss dependent on the training samples and their corresponding
responses, $\mathbf{W}=[\mathbf{w}_{1}^{\rm T},\mathbf{w}_{2}^{\rm
T},\ldots,\mathbf{w}_{s}^{\rm T}]^{\rm T}$ is divided into $s$ non-overlapping
groups, $\mathbf{w}_{i}\in\mathbb{R}^{p_{i}},i=1,2.\ldots,s$, $\lambda>0$ is
the regularization parameter, and

$\varpi(\mathbf{W})=\sum_{i=1}^{s}\|\mathbf{w}_{i}\|_{q}$ (2)

is the $\ell_{1}/\ell_{q}$ norm with $\|\cdot\|_{q}$ denoting the vector
$\ell_{q}$ norm ($q\geq 1$). The $\ell_{1}/\ell_{q}$-regularization belongs to
the composite absolute penalties (CAP) [38] family. When $q=1$, the problem
(1) reduces to the $\ell_{1}$-regularized problem. When $q>1$, the
$\ell_{1}/\ell_{q}$-regularization facilitates group sparsity in the resulting
model, which is desirable in many applications of regression and
classification.

The practical challenge in the use of the $\ell_{1}/\ell_{q}$-regularization
lies in the development of efficient algorithms for solving (1), due to the
non-smoothness of the $\ell_{1}/\ell_{q}$-regularization. According to the
black-box Complexity Theory [25, 26], the optimal first-order black-box method
[25, 26] for solving the class of nonsmooth convex problems converges as
$O(\frac{1}{\sqrt{k}})$ ($k$ denotes the number of iterations), which is slow.
Existing algorithms focus on solving the problem (1) or its equivalent
constrained version for $q=2,\infty$, and they can not be easily extended to
the general case. In order to systematically study the practical performance
of the $\ell_{1}/\ell_{q}$-regularization family, it is of great importance to
develop efficient algorithms for solving (1) for any $q$ larger than $1$.

###  1.1 First-Order Methods Applicable for (1)

When treating $f(\cdot)$ as the general non-smooth convex function, we can
apply the subgradient descent [5, 25, 26]:

$\mathbf{X}_{i+1}=\mathbf{X}_{i}-\gamma_{i}\mathbf{G}_{i},$ (3)

where $\mathbf{G}_{i}\in\partial f(\mathbf{X}_{i})$ is a subgradient of
$f(\cdot)$ at $\mathbf{X}_{i}$, and $\gamma_{i}$ a step size. There are
several different types of step size rules, and more details can be found in
[5, 25]. Subgradient descent is proven to converge, and it can yield a
convergence rate of $O(1/\sqrt{k})$ for $k$ iterations. However, SD has the
following two disadvantages: 1) SD converges slowly; and 2) the iterates of SD
are very rarely at the points of non-differentiability [7], thus it might not
achieve the desirable sparse solution (which is usually at the point of non-
differentiability) within a limited number of iterations.

Coordinate Descent [33] and its recent extension—Coordinate Gradient Descent
(CGD) can be applied for optimizing the non-differentiable composite function
[34]. Coordinate descent has been applied for the $\ell_{1}$-norm regularized
least squares [9], $\ell_{1}/\ell_{\infty}$-norm regularized least squares
[16], and the sparse group Lasso [10]. Coordinate gradient descent has been
applied for the group Lasso logistic regression [21]. Convergence results for
CD and CGD have been established, when the non-differentiable part is
separable [33, 34]. However, there is no global convergence rate for CD and
CGD (Note, CGD is reported to have a _local_ linear convergence rate under
certain conditions [34, Theorem 4]). In addition, it is not clear whether CD
and CGD are applicable for solving the problem (1) with an arbitrary $q\geq
1$.

Fixed Point Continuation [12, 31] was recently proposed for solving the
$\ell_{1}$-norm regularized optimization (i.e.,
$\varpi(\mathbf{W})=\|\mathbf{W}\|_{1}$). It is based on the following fixed
point iteration:

$\mathbf{X}_{i+1}=\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{X}_{i}-\tau
l^{\prime}(\mathbf{X}_{i})),$ (4)

where
$\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{W})=\mbox{sgn}(\mathbf{W})\odot\max(\mathbf{W}-\lambda\tau,0)$
is an operator and $\tau>0$ is the step size. The fixed point iteration (4)
can be applied to solve (1) for any convex penalty $\varpi(\mathbf{W})$, with
the operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$ being defined as:

$\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{W})=\arg\min_{\mathbf{X}}\frac{1}{2}\|\mathbf{X}-\mathbf{W}\|_{2}^{2}+\lambda\tau\varphi(\mathbf{X}).$
(5)

The operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$ is called the
proximal operator [13, 22, 36], and is guaranteed to be non-expansive. With a
properly chosen $\tau$, the fixed point iteration (4) can converge to the
fixed point $\mathbf{X}^{*}$ satisfying

$\mathbf{X}^{*}=\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{X}^{*}-\tau
l^{\prime}(\mathbf{X}^{*})).$ (6)

It follows from (5) and (6) that,

$\mathbf{0}\in\mathbf{X}^{*}-(\mathbf{X}^{*}-\tau
l^{\prime}(\mathbf{X}^{*}))+\lambda\tau\partial\varpi(\mathbf{X}^{*}),$ (7)

which together with $\tau>0$ indicates that $\mathbf{X}^{*}$ is the optimal
solution to (1). In [3, 27], the gradient descent method is extended to
optimize the composite function in the form of (1), and the iteration step is
similar to (4). The extended gradient descent method is proven to yield the
convergence rate of $O(1/k)$ for $k$ iterations. However, as pointed out in
[3, 27], the scheme in (4) can be further accelerated for solving (1).

Finally, there are various online learning algorithms that have been developed
for dealing with large-scale data, e.g., the truncated gradient method [15],
the forward-looking subgradient [7], and the regularized dual averaging [35]
(which is based on the dual averaging method proposed in [28]). When applying
the aforementioned online learning methods for solving (1), a key building
block is the operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$.

###  1.2 Main Contributions

In this paper, we develop an efficient algorithm for solving the
$\ell_{1}/\ell_{q}$-regularized problem (1), for any $q\geq 1$. More
specifically, we develop the GLEP1q algorithm111GLEP1q stands for Group
Sparsity Learning via the $\ell_{1}/\ell_{q}$-regularized Euclidean
Projection., which makes use of the accelerated gradient method [3, 27] for
minimizing the composite objective functions. GLEP1q has the following two
favorable properties: (1) It is applicable to any smooth convex loss
$l(\cdot)$ (e.g., the least squares loss and the logistic loss) and any $q\geq
1$. Existing algorithms are mainly focused on
$\ell_{1}/\ell_{2}$-regularization and/or
$\ell_{1}/\ell_{\infty}$-regularization. To the best of our knowledge, this is
the first work that provides an efficient algorithm for solving (1) with any
$q\geq 1$; and (2) It achieves a global convergence rate of
$O(\frac{1}{k^{2}})$ ($k$ denotes the number of iterations) for the smooth
convex loss $l(\cdot)$. In comparison, although the methods proposed in [1, 6,
16, 29] converge, there is no known convergence rate; and the method proposed
in [21] has a _local_ linear convergence rate under certain conditions [34,
Theorem 4]. In addition, these methods are not applicable for an arbitrary
$q\geq 1$.

The main technical contribution of this paper is the development of an
efficient algorithm for computing the $\ell_{1}/\ell_{q}$-regularized
Euclidean projection (EP1q), which is a key building block in the proposed
GLEP1q algorithm. More specifically, we analyze the key theoretical properties
of the solution of EP1q, based on which we develop an efficient algorithm for
EP1q by solving two zero finding problems. In addition, our theoretical
analysis reveals why EP1q for the general $q$ is significantly more
challenging than the special cases such as $q=2$. We have conducted
experimental studies to demonstrate the efficiency of the proposed algorithm.

###  1.3 Related Work

We briefly review recent studies on $\ell_{1}/\ell_{q}$-regularization, most
of which focus on $\ell_{1}/\ell_{2}$-regularization and/or
$\ell_{1}/\ell_{\infty}$-regularization.

$\ell_{1}/\ell_{2}$-Regularization: The group Lasso was proposed in [37] to
select the groups of variables for prediction in the least squares regression.
In [21], the idea of group lasso was extended for classification by the
logistic regression model, and an algorithm via the coordinate gradient
descent [34] was developed. In [29], the authors considered joint covariate
selection for grouped classification by the logistic loss, and developed a
blockwise boosting Lasso algorithm with the boosted Lasso [39]. In [1], the
authors proposed to learn the sparse representations shared across multiple
tasks, and designed an alternating algorithm. The Spectral projected-gradient
(Spg) algorithm was proposed for solving the $\ell_{1}/\ell_{2}$-ball
constrained smooth optimization problem [4], equipped with an efficient
Euclidean projection that has expected linear runtime. The
$\ell_{1}/\ell_{2}$-regularized multi-task learning was proposed in [18], and
the equivalent smooth reformulations were solved by the Nesterov’s method
[26].

$\ell_{1}/\ell_{\infty}$-Regularization: A blockwise coordinate descent
algorithm [33] was developed for the mutli-task Lasso [16]. It was applied to
the neural semantic basis discovery problem. In [30], the authors considered
the multi-task learning via the $\ell_{1}/\ell_{\infty}$-regularization, and
proposed to solve the equivalent $\ell_{1}/\ell_{\infty}$-ball constrained
problem by the projected gradient descent. In [24], the authors considered the
multivariate regression via the $\ell_{1}/\ell_{\infty}$-regularization,
showed that the high-dimensional scaling of
$\ell_{1}/\ell_{\infty}$-regularization is qualitatively similar to that of
ordinary $\ell_{1}$-regularization, and revealed that, when the overlap
parameter is large enough ($>2/3$), $\ell_{1}/\ell_{\infty}$-regularization
yields the improved statistical efficiency over $\ell_{1}$-regularization.

$\ell_{1}/\ell_{q}$-Regularization: In [6], the authors studied the problem of
boosting with structural sparsity, and developed several boosting algorithms
for regularization penalties including $\ell_{1}$, $\ell_{\infty}$,
$\ell_{1}/\ell_{2}$, and $\ell_{1}/\ell_{\infty}$. In [38], the composite
absolute penalties (CAP) family was introduced, and an algorithm called iCAP
was developed. iCAP employed the least squares loss and the
$\ell_{1}/\ell_{\infty}$ regularization, and was implemented by the boosted
Lasso [39]. The multivariate regression with the
$\ell_{1}/\ell_{q}$-regularization was studied in [17]. In [23], a unified
framework was provided for establishing consistency and convergence rates for
the regularized $M$-estimators, and the results for $\ell_{1}/\ell_{q}$
regularization was established.

###  1.4 Notation

Throughout this paper, scalars are denoted by italic letters, and vectors by
bold face letters. Let $\mathbf{X},\mathbf{Y},\ldots$ denote the
$p$-dimensional parameters, $\mathbf{x}_{i},\mathbf{y}_{i},\ldots$ the
$p_{i}$-dimensional parameters of the $i$-th group, and $x_{i}$ the $i$-th
component of $\mathbf{x}$. We denote $\bar{q}=\frac{q}{q-1}$, and thus $q$ and
$\bar{q}$ satisfy the following relationship:
$\frac{1}{\bar{q}}+\frac{1}{q}=1$. We use the following componentwise
operators: $\odot$, $|\cdot|$ and ${\rm sgn}(\cdot)$. Specifically,
$\mathbf{z}=\mathbf{x}\odot\mathbf{y}$ denotes $z_{i}=x_{i}y_{i}$;
$\mathbf{y}=|\mathbf{x}|$ denotes $y_{i}=|x_{i}|$; and $\mathbf{y}={\rm
sgn}(\mathbf{x})$ denotes $y_{i}={\rm sgn}(x_{i})$, where ${\rm sgn}(\cdot)$
is the signum function: ${\rm sgn}(t)=1$ if $t>0$; ${\rm sgn}(t)=0$ if $t=0$;
and ${\rm sgn}(t)=-1$ if $t<0$.

##  2 The Proposed GLEP1q Algorithm

In this section, we present the proposed GLEP1q algorithm for solving (1) in
the batch learning setting. The main technical contribution lies in the
development of an efficient algorithm for the $\ell_{1}/\ell_{q}$-regularized
Euclidean projection. Specifically, we analyze the key theoretical properties
of the projection in Section 2.1, and show that the projection can be computed
by solving two zero finding problems in Section 2.2. Note that, one can
develop the online learning algorithm for (1) using the online learning
algorithms discussed in the last section, where the
$\ell_{1}/\ell_{q}$-regularized Euclidean projection is also a key building
block.

We first construct the following model for approximating the composite
function $\mathcal{M}(\cdot)$ at the point $\mathbf{X}$ [3, 27]:

$\displaystyle\mathcal{M}_{L,\mathbf{X}}(\mathbf{Y})=[\mbox{loss}(\mathbf{X})+\langle\mbox{loss}^{\prime}(\mathbf{X}),\mathbf{Y}-\mathbf{X}\rangle]+\lambda\varpi(\mathbf{Y})+\frac{L}{2}\|\mathbf{Y}-\mathbf{X}\|_{2}^{2},$
(8)

where $L>0$. In the model $\mathcal{M}_{L,\mathbf{X}}(\mathbf{Y})$, we apply
the first-order Taylor expansion at the point $\mathbf{X}$ (including all
terms in the square bracket) for the smooth loss function $l(\cdot)$, and
directly put the non-smooth penalty $\varpi(\cdot)$ into the model. The
regularization term $\frac{L}{2}\|\mathbf{Y}-\mathbf{X}\|_{2}^{2}$ prevents
$\mathbf{Y}$ from walking far away from $\mathbf{X}$, thus the model can be a
good approximation to $f(\mathbf{Y})$ in the neighborhood of $\mathbf{X}$.

The accelerated gradient method is based on two sequences
$\\{\mathbf{X}_{i}\\}$ and $\\{\mathbf{S}_{i}\\}$ in which
$\\{\mathbf{X}_{i}\\}$ is the sequence of approximate solutions, and
$\\{\mathbf{S}_{i}\\}$ is the sequence of search points. The search point
$\mathbf{S}_{i}$ is the affine combination of $\mathbf{X}_{i-1}$ and
$\mathbf{X}_{i}$ as

$\mathbf{S}_{i}=\mathbf{X}_{i}+\beta_{i}(\mathbf{X}_{i}-\mathbf{X}_{i-1}),$
(9)

where $\beta_{i}$ is a properly chosen coefficient. The approximate solution
$\mathbf{X}_{i+1}$ is computed as the minimizer of
$\mathcal{M}_{L_{i},\mathbf{S}_{i}}(\mathbf{Y})$:

$\mathbf{X}_{i+1}=\arg\min_{\mathbf{Y}}\mathcal{M}_{L_{i},\mathbf{S}_{i}}(\mathbf{Y}),$
(10)

where $L_{i}$ is determined by line search, e.g., the Armijo-Goldstein rule so
that $L_{i}$ should be appropriate for $\mathbf{S}_{i}$.

Algorithm 1 GLEP1q: Group Sparsity Learning via the
$\ell_{1}/\ell_{q}$-regularized Euclidean Projection

0: $\lambda_{1}\geq 0,\lambda_{2}\geq 0,L_{0}>0,\mathbf{X}_{0},k$

0: $\mathbf{X}_{k+1}$

1: Initialize $\mathbf{X}_{1}=\mathbf{X}_{0}$, $\alpha_{-1}=0$,
$\alpha_{0}=1$, and $L=L_{0}$.

2: for $i=1$ to $k$ do

3: Set $\beta_{i}=\frac{\alpha_{i-2}-1}{\alpha_{i-1}}$,
$\mathbf{S}_{i}=\mathbf{X}_{i}+\beta_{i}(\mathbf{X}_{i}-\mathbf{X}_{i-1})$

4: Find the smallest $L=L_{i-1},2L_{i-1},\ldots$ such that
$f(\mathbf{X}_{i+1})\leq\mathcal{M}_{L,\mathbf{S}_{i}}(\mathbf{X}_{i+1}),$
where
$\mathbf{X}_{i+1}=\arg\min_{\mathbf{Y}}\mathcal{M}_{L,\mathbf{S}_{i}}(\mathbf{Y})$

5: Set $L_{i}=L$ and $\alpha_{i+1}=\frac{1+\sqrt{1+4\alpha_{i}^{2}}}{2}$

6: end for

The algorithm for solving (1) is presented in Algorithm 1. GLEP1q inherits the
optimal convergence rate of $O(1/k^{2})$ from the accelerated gradient method.
In Algorithm 1, a key subroutine is (10), which can be computed as
$\mathbf{X}_{i+1}=\pi_{1q}(\mathbf{S}_{i}-l^{\prime}(\mathbf{S}_{i})/L_{i},\lambda/L_{i})$,
where $\pi_{1q}(\cdot)$ is the $\ell_{1}/\ell_{q}$-regularized Euclidean
projection (EP1q) problem:

$\pi_{1q}(\mathbf{V},\lambda)=\arg\min_{\mathbf{X}\in\mathbb{R}^{p}}\frac{1}{2}\|\mathbf{X}-\mathbf{V}\|_{2}^{2}+\lambda\sum_{i=1}^{s}\|\mathbf{x}_{i}\|_{q}.$
(11)

The efficient computation of (11) for any $q>1$ is the main technical
contribution of this paper. Note that the $s$ groups in (11) are independent.
Thus the optimization in (11) decouples into a set of $s$ independent
$\ell_{q}$-regularized Euclidean projection problems:

$\pi_{q}(\mathbf{v})=\arg\min_{\mathbf{x}\in\mathbb{R}^{n}}\left(g(\mathbf{x})=\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{q}\right),$
(12)

where $n=p_{i}$ for the $i$-th group. Next, we study the key properties of
(12).

###  2.1 Properties of the Optimal Solution to (12)

The function $g(\cdot)$ is strictly convex, and thus it has a unique
minimizer, as summarized below:

###### Lemma 1

The problem (12) has a unique minimizer.

Next, we show that the optimal solution to (12) is given by zero under a
certain condition, as summarized in the following theorem:

###### Theorem 1

$\pi_{q}(\mathbf{v})=\mathbf{0}$ if and only if
$\lambda\geq\|\mathbf{v}\|_{\bar{q}}$.

Proof:  Let us first compute the directional derivative of $g(\mathbf{x})$ at
the point $\mathbf{0}$:

$Dg(\mathbf{0})[\mathbf{u}]=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}[g(\alpha\mathbf{u})-g(\mathbf{0})]=-\langle\mathbf{v},\mathbf{u}\rangle+\lambda\|\mathbf{u}\|_{q},$

where $\mathbf{u}$ is a given direction. According to the Hölder’s inequality,
we have

$|\langle\mathbf{u},\mathbf{v}\rangle|\leq\|\mathbf{u}\|_{q}\|\mathbf{v}\|_{\bar{q}},\forall\mathbf{u}.$

Therefore, we have

$Dg(\mathbf{0})[\mathbf{u}]\geq 0,\forall\mathbf{u},$ (13)

if and only if $\lambda\geq\|\mathbf{v}\|_{\bar{q}}$. The result follows,
since (13) is the necessary and sufficient condition for $\mathbf{0}$ to be
the optimal solution of (12). $\Box$

Next, we focus on solving (12) for $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. We
first consider solving (12) in the case of $1<q<\infty$, which is the main
technical contribution of this paper. We begin with a lemma that summarizes
the key properties of the optimal solution to the problem (12):

###### Lemma 2

Let $1<q<\infty$ and $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. Then,
$\mathbf{x}^{*}$ is the optimal solution to the problem (12) if and if only it
satisfies:

$\mathbf{x}^{*}+\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}{\mathbf{x}^{*}}^{(q-1)}=\mathbf{v},$
(14)

where $\mathbf{y}\equiv\mathbf{x}^{(q-1)}$ is defined component-wisely as:
$y_{i}={\rm sgn}(x_{i})|x_{i}|^{q-1}$. Moreover, we have

$\pi_{q}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\pi_{q}(|\mathbf{v}|),$ (15)
${\rm sgn}(\mathbf{x}^{*})={\rm sgn}(\mathbf{v}),$ (16)
$0<|x^{*}_{i}|<|v_{i}|,\forall i\in\\{i|v_{i}\neq 0\\}.$ (17)

Proof:  Since $\lambda<\|\mathbf{v}\|_{\bar{q}}$, it follows from Theorem 1
that the optimal solution $\mathbf{x}^{*}\neq\mathbf{0}$. $\|\mathbf{x}\|_{q}$
is differentiable when $\mathbf{x}\neq\mathbf{0}$, so is $g(\mathbf{x})$.
Therefore, the sufficient and necessary condition for $\mathbf{x}^{*}$ to be
the solution of (12) is $g^{\prime}(\mathbf{x}^{*})=0$, i.e., (14). Denote
$c^{*}\equiv\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}>0$. It follows from (14) that
(15) holds, and

$\mbox{sgn}(x^{*}_{i})\left(|x^{*}_{i}|+c^{*}|x^{*}_{i}|^{q-1}\right)=v_{i},$
(18)

from which we can verify (16) and (17). $\Box$

Figure 1: Illustration of the failure of the fixed point iteration
$\mathbf{x}=\mathbf{v}-\lambda\|\mathbf{x}\|_{q}^{1-q}{\mathbf{x}}^{(q-1)}$
for solving (12). We set $\mathbf{v}=[1,3]^{\rm T}$ and the starting point
$\mathbf{x}=[1,3]^{\rm T}$. The vertical axis denotes the values of $x_{1}$
during the iterations.

It follows from Lemma 2 that i) if $v_{i}=0$ then $x_{i}^{*}=0$; and ii)
$\pi_{q}(\mathbf{v})$ can be easily obtained from $\pi_{q}(|\mathbf{v}|)$.
Thus, we can restrict our following discussion to $\mathbf{v}>\mathbf{0}$,
i.e., $v_{i}>0,\forall i$. It is clear that, the analysis can be easily
extended to the general $\mathbf{v}$. The optimality condition in (14)
indicates that $\mathbf{x}^{*}$ might be solved via the fixed point iteration

$\mathbf{x}=\eta(\mathbf{x})\equiv\mathbf{v}-\lambda\|\mathbf{x}\|_{q}^{1-q}{\mathbf{x}}^{(q-1)},$

which is, however, not guaranteed to converge (see Figure 1 for examples), as
$\eta(\cdot)$ is not necessarily a contraction mapping [14, Proposition 3]. In
addition, $\mathbf{x}^{*}$ cannot be trivially solved by firstly guessing
$c=\|\mathbf{x}\|_{q}^{1-q}$ and then finding the root of $\mathbf{x}+\lambda
c{\mathbf{x}}^{(q-1)}=\mathbf{v}$, as when $c$ increases, the values of
$\mathbf{x}$ obtained from $\mathbf{x}+\lambda
c{\mathbf{x}}^{(q-1)}=\mathbf{v}$ decrease, so that
$c=\|\mathbf{x}\|_{q}^{1-q}$ increases as well (note that $1-q<0$).

###  2.2 Computing the Optimal Solution $\mathbf{x}^{*}$ by Zero Finding

In the following, we show that $\mathbf{x}^{*}$ can be obtained by solving two
zero finding problems. Below, we construct our first auxiliary function
$h_{c}^{v}(\cdot)$ and reveal its properties:

######  Definition 1 (Auxiliary Function $h_{c}^{v}(\cdot)$ )

Let $c>0$, $1<q<\infty$, and $v>0$. We define the auxiliary function
$h_{c}^{v}(\cdot)$ as follows:

$h_{c}^{v}(x)=x+cx^{q-1}-v,0\leq x\leq v.$ (19)

###### Lemma 3

Let $c>0$, $1<q<\infty$, and $v>0$. Then, $h_{c}^{v}(\cdot)$ has a unique root
in the interval $(0,v)$.

Proof:  It is clear that $h_{c}^{v}(\cdot)$ is continuous and strictly
increasing in the interval $[0,v]$, $h_{c}^{v}(0)=-v<0$, and
$h_{c}^{v}(v)=cv^{q-1}>0$. According to the Intermediate Value Theorem,
$h_{c}^{v}(\cdot)$ has a unique root lying in the interval $(0,v)$. This
concludes the proof. $\Box$

###### Corolary 1

Let $\mathbf{x},\mathbf{v}\in\mathbb{R}^{n}$, $c>0$, $1<p<\infty$, and
$\mathbf{v}>\mathbf{0}$. Then, the function

$\varphi_{c}^{\mathbf{v}}(\mathbf{x})=\mathbf{x}+c\mathbf{x}^{(q-1)}-\mathbf{v},\mathbf{0}<\mathbf{x}<\mathbf{v}$
(20)

has a unique root.

Let $\mathbf{x}^{*}$ be the optimal solution satisfying (14). Denote
$c^{*}=\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}$. It follows from Lemma 2 and
Corollary 1 that $\mathbf{x}^{*}$ is the unique root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ defined in (20), provided that the
optimal $c^{*}$ is known. Our methodology for computing $\mathbf{x}^{*}$ is to
first compute the optimal $c^{*}$ and then compute $\mathbf{x}^{*}$ by
computing the root of $\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$. Next, we show how
to compute the optimal $c^{*}$ by solving a single variable zero finding
problem. We need our second auxiliary function $\omega(\cdot)$ defined as
follows:

######  Definition 2 (Auxiliary Function $\omega(\cdot)$)

Let $1<q<\infty$ and $v>0$. We define the auxiliary function $\omega(\cdot)$
as follows:

$c=\omega(x)=(v-x)/x^{q-1},0<x\leq v.$ (21)

###### Lemma 4

In the interval $(0,v]$, $c=\omega(x)$ is i) continuously differentiable, ii)
strictly decreasing, and iii) invertible. Moreover, in the domain
$[0,\infty)$, the inverse function $x=\omega^{-1}(c)$ is continuously
differentiable and strictly decreasing.

Proof:  It is easy to verify that, in the interval $(0,v]$, $c=\omega(x)$ is
continuously differentiable with a non-positive gradient, i.e.,
$\omega^{\prime}(x)<0$. Therefore, the results follow from the Inverse
Function Theorem. $\Box$

It follows from Lemma 4 that given the optimal $c^{*}$ and $\mathbf{v}$, the
optimal $\mathbf{x}^{*}$ can be computed via the inverse function
$\omega^{-1}(\cdot)$, i.e., we can represent $\mathbf{x}^{*}$ as a function of
$c^{*}$. Since $\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}-c^{*}=0$ by the definition
of $c^{*}$, the optimal $c^{*}$ is a root of our third auxiliary function
$\phi(\cdot)$ defined as follows:

######  Definition 3 (Auxiliary Function $\phi(\cdot)$)

Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, and
$\mathbf{v}>\mathbf{0}$. We define the auxiliary function $\phi(\cdot)$ as
follows:

$\phi(c)=\lambda\psi(c)-c,c\geq 0,$ (22)

where

$\psi(c)=\left(\sum_{i=1}^{n}(\omega_{i}^{-1}(c))^{q}\right)^{\frac{1-q}{q}},$
(23)

and $\omega_{i}^{-1}(c)$ is the inverse function of

$\omega_{i}(x)=(v_{i}-x)/x^{q-1},0<x\leq v_{i}.$ (24)

Recall that we assume $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$ (otherwise the
optimal solution is given by zero from Theorem 1). The following lemma
summarizes the key properties of the auxiliary function $\phi(\cdot)$:

###### Lemma 5

Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, and

$\epsilon=(\|\mathbf{v}\|_{\bar{q}}-\lambda)/\|\mathbf{v}\|_{\bar{q}}.$ (25)

Then, $\phi(\cdot)$ is continuously differentiable in the interval
$[0,\infty)$. Moreover, we have

$\phi(0)=\lambda\|\mathbf{v}\|_{q}^{1-q}>0,\phi(\overline{c})\leq 0,$

where

$\overline{c}=\max_{i}c_{i},$ (26)
$c_{i}=\omega_{i}(v_{i}\epsilon),i=1,2,\ldots,n.$ (27)

Proof:  From Lemma 4, the function $\omega_{i}^{-1}(c)$ is continuously
differentiable in $[0,\infty)$. It is easy to verify that
$\omega_{i}^{-1}(c)>0,\forall c\in[0,\infty)$. Thus, $\phi(\cdot)$ in (22) is
continuously differentiable in $[0,\infty)$.

It is clear that $\phi(0)=\lambda\|\mathbf{v}\|_{q}^{1-q}>0$. Next, we show
$\phi(\overline{c})\leq 0$. Since $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, we
have

$0<\epsilon<1.$ (28)

It follows from (24), (26), (27) and (28) that
$0<c_{i}\leq\overline{c},\forall i$. Let
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\rm T}$ be the root of
$\varphi_{\overline{c}}^{\mathbf{v}}(\cdot)$ (see Corollary 1). Then,
$x_{i}=\omega^{-1}_{i}(\overline{c})$. Since $\omega^{-1}_{i}(\cdot)$ is
strictly decreasing (see Lemma 4), $c_{i}\leq\overline{c}$,
$v_{i}\epsilon=\omega_{i}^{-1}(c_{i})$, and
$x_{i}=\omega_{i}^{-1}(\overline{c})$, we have

$x_{i}\leq v_{i}\epsilon.$ (29)

Combining (24), (29), and $\overline{c}=\omega_{i}(x_{i})$, we have
$\overline{c}\geq v_{i}(1-\epsilon)/x_{i}^{q-1}$, since $\omega_{i}(\cdot)$ is
strictly decreasing. It follows that
$x_{i}\geq\left(\frac{v_{i}(1-\epsilon)}{\overline{c}}\right)^{\frac{1}{q-1}}$.
Thus, the following holds:

$\psi(\overline{c})=\left(\sum_{i=1}^{n}(\omega_{i}^{-1}(\overline{c}))^{q}\right)^{\frac{1-q}{q}}=\left(\sum_{i=1}^{n}x_{i}^{q}\right)^{\frac{1-q}{q}}\leq\frac{\overline{c}}{\|\mathbf{v}\|_{\bar{q}}(1-\epsilon)},$

which leads to

$\phi(\overline{c})=\lambda\psi(\overline{c})-\overline{c}\leq\overline{c}\left(\frac{\lambda}{\|\mathbf{v}\|_{\bar{q}}(1-\epsilon)}-1\right)=0,$

where the last equality follows from (25). $\Box$

###### Corolary 2

Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, and $\underline{c}=\min_{i}c_{i}$, where $c_{i}$’s
are defined in (27). We have $0<\underline{c}\leq\overline{c}$ and
$\phi(\underline{c})\geq 0$.

Following Lemma 5 and Corollary 2, we can find at least one root of
$\phi(\cdot)$ in the interval $[\underline{c},\overline{c}]$. In the following
theorem, we show that $\phi(\cdot)$ has a unique root:

###### Theorem 2

Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, and
$\mathbf{v}>\mathbf{0}$. Then, in $[\underline{c},\overline{c}]$,
$\phi(\cdot)$ has a unique root, denoted by $c^{*}$, and the root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ is the optimal solution to (12).

Proof:  From Lemma 5 and Corollary 2, we have $\phi(\overline{c})\leq 0$ and
$\phi(\underline{c})\geq 0$. If either $\phi(\overline{c})=0$ or
$\phi(\underline{c})=0$, $\overline{c}$ or $\underline{c}$ is a root of
$\phi(\cdot)$. Otherwise, we have $\phi(\underline{c})\phi(\overline{c})<0$.
As $\phi(\cdot)$ is continuous in $[0,\infty)$, we conclude that $\phi(\cdot)$
has a root in $(\underline{c},\overline{c})$ according to the Intermediate
Value Theorem.

Next, we show that $\phi(\cdot)$ has a unique root in the interval
$[0,\infty)$. We prove this by contradiction. Assume that $\phi(\cdot)$ has
two roots: $0<c_{1}<c_{2}$. From Corollary 1,
$\varphi_{c_{1}}^{\mathbf{v}}(\cdot)$ and
$\varphi_{c_{2}}^{\mathbf{v}}(\cdot)$ have unique roots. Denote
$\mathbf{x}^{1}=[x_{1}^{1},x_{2}^{1},\ldots,x_{n}^{1}]^{\rm T}$ and
$\mathbf{x}^{2}=[x_{1}^{2},x_{2}^{2},\ldots,x_{n}^{2}]^{\rm T}$ as the roots
of $\varphi_{c_{1}}^{\mathbf{v}}(\cdot)$ and
$\varphi_{c_{2}}^{\mathbf{v}}(\cdot)$, respectively. We have
$0<x_{i}^{1},x_{i}^{2}<v_{i},\forall i$. It follows from (22-24) that

$\displaystyle\mathbf{x}^{1}+\lambda\|\mathbf{x}^{1}\|_{q}^{1-q}{\mathbf{x}^{1}}^{(q-1)}-\mathbf{v}=\mathbf{0},$
$\displaystyle\mathbf{x}^{2}+\lambda\|\mathbf{x}^{2}\|_{q}^{1-q}{\mathbf{x}^{2}}^{(q-1)}-\mathbf{v}=\mathbf{0}.$

According to Lemma 2, $\mathbf{x}^{1}$ and $\mathbf{x}^{2}$ are the optimal
solution of (12). From Lemma 1, we have $\mathbf{x}^{1}=\mathbf{x}^{2}$.
However, since $x_{i}^{1}=\omega_{i}^{-1}(c_{1})$,
$x_{i}^{2}=\omega_{i}^{-1}(c_{2})$, $\omega_{i}^{-1}(\cdot)$ is a strictly
decreasing function in $[0,\infty)$ by Lemma 4, and $c_{1}<c_{2}$, we have
$x_{i}^{1}>x_{i}^{2},\forall i$. This leads to a contradiction. Therefore, we
conclude that $\phi(\cdot)$ has a unique root in
$[\underline{c},\overline{c}]$.

From the above arguments, it is clear that, the root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ is the optimal solution to (12). $\Box$

###### Remark 1

When $q=2$, we have
$\underline{c}=\overline{c}=\frac{\lambda}{\|\mathbf{v}\|_{2}-\lambda}$. It is
easy to verify that $\phi(\underline{c})=\phi(\overline{c})=0$ and

$\pi_{2}(\mathbf{v})=\frac{\|\mathbf{v}\|_{2}-\lambda}{\|\mathbf{v}\|_{2}}\mathbf{v}.$
(30)

Therefore, when $q=2$, we obtain a closed-form solution.

###  2.3 Solving the Zero Finding Problem by Bisection

Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, $\overline{v}=\max_{i}v_{i}$,
$\underline{v}=\min_{i}v_{i}$, and $\delta>0$ be a small constant (e.g.,
$\delta=10^{-8}$ in our experiments). When $q>2$, we have

$\underline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\overline{v}^{q-2}}\quad\mbox{
and }\quad\overline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\underline{v}^{q-2}}.$

When $1<q<2$, we have

$\underline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\underline{v}^{q-2}}\quad\mbox{
and }\quad\overline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\overline{v}^{q-2}}.$

If either $\phi(\overline{c})=0$ or $\phi(\underline{c})=0$, $\overline{c}$ or
$\underline{c}$ is the unique root of $\phi(\cdot)$. Otherwise, we can find
the unique root of $\phi(\cdot)$ by bisection in the interval
$(\underline{c},\overline{c})$, which costs at most

$N=\log_{2}\frac{(1-\epsilon)|\overline{v}^{q-2}-\underline{v}^{q-2}|}{\epsilon^{q-1}\overline{v}^{q-2}\underline{v}^{q-2}\delta}$

iterations for achieving an accuracy of $\delta$. Let $[c_{1},c_{2}]$ be the
current interval of uncertainty, and we have computed $\omega_{i}^{-1}(c_{1})$
and $\omega_{i}^{-1}(c_{2})$ in the previous bisection iterations. Setting
$c=\frac{c_{1}+c_{2}}{2}$, we need to evaluate $\phi(c)$ by computing
$\omega_{i}^{-1}(c),i=1,2,\ldots,n$. It is easy to verify that
$\omega_{i}^{-1}(c)$ is the root of $h_{c}^{v_{i}}(\cdot)$ in the interval
$(0,v_{i})$. Since $\omega_{i}^{-1}(\cdot)$ is a strictly decreasing function
(see Lemma 4), the following holds:

$\omega_{i}^{-1}(c_{2})<\omega_{i}^{-1}(c)<\omega_{i}^{-1}(c_{1}),$

and thus $\omega_{i}^{-1}(c)$ can be solved by bisection using at most

$\log_{2}\frac{\omega_{i}^{-1}(c_{2})-\omega_{i}^{-1}(c_{1})}{\delta}<\log_{2}\frac{v_{i}}{\delta}\leq\log_{2}\frac{\overline{v}}{\delta}$

iterations for achieving an accuracy of $\delta$. For given
$\mathbf{v},\lambda$, and $\delta$, $N$ and $\overline{v}$ are constant, and
thus it costs $O(n)$ for finding the root of $\phi(\cdot)$. Once $c^{*}$, the
root of $\phi(\cdot)$ is found, it costs $O(n)$ flops to compute
$\mathbf{x}^{*}$ as the unique root of $\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$.
Therefore, the overall time complexity for solving (12) is $O(n)$.

We have shown how to solve (12) for $1<q<\infty$. For $q=1$, the problem (12)
is reduced to the one used in the standard Lasso, and it has the following
closed-form solution [3]:

$\pi_{1}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-\lambda,0).$
(31)

For $q=\infty$, the problem (12) can computed via (31), as summarized in the
following theorem:

###### Theorem 3

Let $q=\infty$, $\bar{q}=1$, and $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. Then we
have

$\pi_{\infty}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\min(|\mathbf{v}|,t^{*}),$
(32)

where $t^{*}$ is the unique root of

$h(t)=\sum_{i=1}^{n}\max(|v_{i}|-t,0)-\lambda.$ (33)

Proof:  Making use of the property that
$\|\mathbf{x}\|_{\infty}=\max_{\|\mathbf{y}\|_{1}\leq
1}\langle\mathbf{y},\mathbf{x}\rangle$, we can rewrite (12) in the case of
$q=\infty$ as

$\min_{\mathbf{x}}\max_{\mathbf{y}:\|\mathbf{y}\|_{1}\leq\lambda}s(\mathbf{x},\mathbf{y})\equiv\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\langle\mathbf{y},\mathbf{x}\rangle.$
(34)

The function $s(\mathbf{x},\mathbf{y})$ is continuously differentiable in both
$\mathbf{x}$ and $\mathbf{y}$, convex in $\mathbf{x}$ and concave in
$\mathbf{y}$, and the feasible domains are solids. According to the well-known
von Neumann Lemma [25], the min-max problem (34) has a saddle point, and thus
the minimization and maximization can be exchanged. Setting the derivative of
$s(\mathbf{x},\mathbf{y})$ with respect to $\mathbf{x}$ to zero, we have

$\mathbf{x}=\mathbf{v}-\mathbf{y}.$ (35)

Thus we obtain the following problem:

$\min_{\mathbf{y}:\|\mathbf{y}\|_{1}\leq\lambda}\frac{1}{2}\|\mathbf{y}-\mathbf{v}\|_{2}^{2},$
(36)

which is the problem of the Euclidean projection onto the $\ell_{1}$ ball [4,
6, 20]. It has been shown that the optimal solution $\mathbf{y}^{*}$to (36)
for $\lambda<\|\mathbf{v}\|_{1}$ can be obtained by first computing $t^{*}$ as
the unique root of (33) in linear time, and then computing $\mathbf{y}^{*}$ as

$\mathbf{y}^{*}={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-t^{*},0).$ (37)

It follows from (35) and (37) that (32) holds. $\Box$

We conclude this section by summarizing the main steps for solving the
$\ell_{q}$-regularized Euclidean projection in Algorithm 2.

Algorithm 2 Epq: $\ell_{q}$-regularized Euclidean projection

0: $\lambda>0,q\geq 1,\mathbf{v}\in\mathbb{R}^{n}$

0:
$\mathbf{x}^{*}=\pi_{q}(\mathbf{v})=\arg\min_{\mathbf{x}\in\mathbb{R}^{n}}\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{q}$

1: Compute $\bar{q}=\frac{q}{q-1}$

2: if $\|\mathbf{v}\|_{\bar{q}}\leq\lambda$ then

3: Set $\mathbf{x}^{*}=\mathbf{0}$, return

4: end if

5: if $q=1$ then

6: Set $\mathbf{x}^{*}={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-\lambda,0)$

7: else if $q=2$ then

8: Set
$\mathbf{x}^{*}=\frac{\|\mathbf{v}\|_{2}-\lambda}{\|\mathbf{v}\|_{2}}\mathbf{v}$

9: else if $q=\infty$ then

10: Obtain $t^{*}$, the unique root of $h(t)$, via the improved bisection
method [20]

11: Set $\mathbf{x}^{*}={\rm sgn}(\mathbf{v})\odot\min(|\mathbf{v}|,t^{*})$

12: else

13: Compute $c^{*}$, the unique root of $\phi(c)$, via bisection in the
interval $[\underline{c},\overline{c}]$ (Theorem 2)

14: Obtain $\mathbf{x}^{*}$ as the unique root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$

15: end if

##  3 Experiments

We have conducted experiments to evaluate the efficiency of the proposed
algorithm using both synthetic and real-world data. We set the regularization
parameter as $\lambda=r\times\lambda_{\max}^{q}$, where $0<r\leq 1$ is the
ratio, and $\lambda_{\max}^{q}$ is the maximal value above which the
$\ell_{1}/\ell_{q}$-norm regularized problem (1) obtains a zero solution (see
Theorem 1). We try the following values for $q$: $1.25,1.5,1.75,2,2.33,3,5$,
and $\infty$. The source codes, included in the SLEP package [19], are
available online222http://www.public.asu.edu/~jye02/Software/SLEP/.

###  3.1 Simulation Studies

We use the synthetic data to study the effectiveness of the
$\ell_{1}/\ell_{q}$-norm regularization for reconstructing the jointly sparse
matrix under different values of $q>1$. Let $A\in\mathbb{R}^{m\times d}$ be a
measurement matrix with entries being generated randomly from the standard
normal distribution, $X^{*}\in\mathbb{R}^{d\times k}$ be the jointly sparse
matrix with the first $\tilde{d}<d$ rows being nonzero and the remaining rows
exactly zero, $Y=AX^{*}+Z$ be the response matrix, and
$Z\in\mathbb{R}^{m\times k}$ is the noise matrix whose entries are drawn
randomly from the normal distribution with mean zero and standard deviation
$\sigma=0.1$. We treat each row of $X^{*}$ as a group, and estimate $X^{*}$
from $A$ and $Y$ by solving the following $\ell_{1}/\ell_{q}$-norm regularized
problem:

$X=\arg\min_{W}\frac{1}{2}\|AW-Y\|_{F}^{2}+\lambda\sum_{i=1}^{d}\|W^{i}\|_{q},$

where $W^{i}$ denotes the $i$-th row of $W$. We set $m=100$, $d=200$, and
$\tilde{d}=k=50$. We try two different settings for $X^{*}$, by drawing its
nonzero entries randomly from 1) the uniform distribution in the interval
$[0,1]$ and 2) the standard normal distribution.

Figure 2:  Performance of the $\ell_{1}/\ell_{q}$-norm regularization for
reconstructing the jointly sparse $X^{*}$. The nonzero entries of $X^{*}$ are
drawn randomly from the uniform distribution for the plots in the first row,
and from the normal distribution for the plots in the second row. Plots in the
first two rows show $\|X-X^{*}\|_{F}$, the Frobenius norm difference between
the solution and the truth; and plots in the third row show the
$\ell_{2}$-norm of each row of the solution $X$.

We compute the solutions corresponding to a sequence of decreasing values of
$\lambda=r\times\lambda_{\max}^{q}$, where $r=0.9^{i-1}$, for
$i=1,2,\ldots,100$. In addition, we use the solution corresponding to the
$0.9^{i}\times\lambda_{\max}^{q}$ as the “warm” start for
$0.9^{i+1}\times\lambda_{\max}^{q}$. We report the results in Figure 2, from
which we can observe: 1) the distance between the solution $X$ and the truth
$X^{*}$ usually decreases with decreasing values of $\lambda$; 2) for the
uniform distribution (see the plots in the first row), $q=1.5$ performs the
best; 3) for the normal distribution (see the plots in the second row),
$q=1.5,1.75,2$ and 3 achieve comparable performance and perform better than
$q=1.25$, 5 and $\infty$; 4) with a properly chosen threshold, the support of
$X^{*}$ can be exactly recovered by the $\ell_{1}/\ell_{q}$-norm
regularization with an appropriate value of $q$, e.g., $q=1.5$ for the uniform
distribution, and $q=2$ for the normal distribution; and 5) the recovery of
$X^{*}$ with nonzero entries drawn from the normal distribution is easier than
that with entries generated from the uniform distribution.

The existing theoretical results [17, 23] can not tell which $q$ is the best;
and we believe that the optimal $q$ depends on the distribution of $X^{*}$, as
indicated from the above results. Therefore, it is necessary to conduct the
distribution-specific theoretical studies (note that the previous studies
usually make no assumption on $X^{*}$). The proposed GLEP1q algorithm shall
help verify the theoretical results to be established.

###  3.2 Performance on the Letter Data Set

We apply the proposed GLEP1q algorithm for multi-task learning on the Letter
data set [29], which consists of 45,679 samples from 8 default tasks of two-
class classification problems for the handwritten letters: c/e, g/y, m/n, a/g,
i/j, a/o, f/t, h/n. The writings were collected from over 180 different
writers, with the letters being represented by $8\times 16$ binary pixel
images. We use the least squares loss for $l(\cdot)$.

Figure 3: Computational time (seconds) comparison between GLEP1q ($q=2$) and
Spg under different values of $\lambda=r\times\lambda_{\max}^{q}$ and $m$.

####  3.2.1 Efficiency Comparison with Spg

We compare GLEP1q with the Spg algorithm proposed in [4]. Spg is a specialized
solver for the $\ell_{1}/\ell_{2}$-ball constrained optimization problem, and
has been shown to outperform existing algorithms based on blockwise coordinate
descent and projected gradient. In Figure 3, we report the computational time
under different values of $m$ (the number of samples) and
$\lambda=r\times\lambda_{\max}^{q}$ ($q=2$). It is clear from the plots that
GLEP1q is much more efficient than Spg, which may attribute to: 1) GLEP1q has
a better convergence rate than Spg; and 2) when $q=2$, the EP1q in GLEP1q can
be computed analytically (see Remark 1), while this is not the case in Spg.

####  3.2.2 Efficiency under Different Values of $q$

We report the computational time (seconds) of GLEP1q under different values of
$q$, $\lambda=r\times\lambda_{\max}^{q}$ and $m$ (the number of samples) in
Figure 4. We can observe from this figure that the computational time of
GLEP1q under different values of $q$ (for fixed $r$ and $m$) is comparable.
Together with the result on the comparison with Spg for $q=2$, this experiment
shows the promise of GLEP1q for solving large-scale problems for any $q\geq
1$.

Figure 4: Computation time (seconds) of GLEP1q under different values of $m$,
$q$ and $r$.

Figure 5: The balanced error rate achieved by the $\ell_{1}/\ell_{q}$
regularization under different values of $q$. The title of each plot indicates
the percentages of samples used for training, validation, and testing.

####  3.2.3 Performance under Different Values of $q$

We randomly divide the Letter data into three non-overlapping sets: training,
validation, and testing. We train the model using the training set, and tune
the regularization parameter $\lambda=r\times\lambda_{\max}^{q}$ on the
validation set, where $r$ is chosen from $\\{10^{-1},5\times 10^{-2},2\times
10^{-2},1\times 10^{-2},5\times 10^{-3},2\times 10^{-3},1\times 10^{-3}\\}$.
On the testing set, we compute the balanced error rate [11]. We report the
results averaged over 10 runs in Figure 5. The title of each plot indicates
the percentages of samples used for training, validation, and testing. The
results show that, on this data set, a smaller value of $q$ achieves better
performance.

##  4 Conclusion

In this paper, we propose the GLEP1q algorithm for solving the
$\ell_{1}/\ell_{q}$-norm regularized problem, for any $q\geq 1$. The main
technical contribution of this paper is the efficient algorithm for the
$\ell_{1}/\ell_{q}$-norm regularized Euclidean projection (EP1q), which is a
key building block of GLEP1q. Specifically, we analyze the key theoretical
properties of the solution of EP1q, based on which we develop an efficient
algorithm for EP1q by solving two zero finding problems. Our analysis also
reveals why EP1q for the general $q$ is significantly more challenging than
the special cases such as $q=2$.

In this paper, we focus on the efficient implementation of the
$\ell_{1}/\ell_{q}$-regularized problem. We plan to study the effectiveness of
the $\ell_{1}/\ell_{q}$ regularization under different values of $q$ for real-
world applications in computer vision and bioinformatics. We also plan to
conduct the distribution-specific [8] theoretical studies for different values
of $q$.

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#
Thermodynamics of the Lee-Wick partners: An alternative approach

Kaushik Bhattacharya$, Suratna Das†
$Department of Physics, Indian Institute of Technology, Kanpur,
Kanpur 208016, India
† Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba, Mumbai 400005, India111Major part of the work has
been carried out while at Theoretical Physics Division, Physical Research
Laboratory, Navrangpura, Ahmedabad 380009, India email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It was pointed out some time ago that there can be two variations in which the
divergences of a quantum field theory can be tamed using the ideas presented
by Lee and Wick. In one variation the Lee-Wick partners of the normal fields
live in an indefinite metric Hilbert space but have positive energy and in the
other variation the Lee-Wick partners can live in a normal Hilbert space but
carry negative energy. Quantum mechanically the two variations mainly differ
in the way the fields are quantized. In this article the second variation of
Lee and Wick’s idea is discussed. Using statistical mechanical methods the
energy density, pressure and entropy density of the negative energy Lee-Wick
fields have been calculated. The results exactly match with the thermodynamic
results of the conventional, positive energy Lee-Wick fields. The result sheds
some light on the second variation of Lee-Wick’s idea. The result seems to say
that the thermodynamics of the theories do not care about the way they are
quantized.

##  1 Introduction

The Lee-Wick field theories [1, 2] originated in an attempt to address the
problem related with the infinities in quantum field theories. Recently some
authors have tried to implement Lee-Wick’s idea in a higher derivative version
of a quantum field theory [3, 4]. All these theories assumed the existence of
some partners of the Standard model particles. The main ideas of the Lee-Wick
Standard Model in Ref. [3] have been extended in Ref. [5] where the authors
use two Lee-Wick partners for each standard model field: one with negative and
the other with positive norm. Later, this idea has been used in Ref. [6] to
improve gauge coupling unification without introducing additional fields in
the higher-derivative theory. The Higgs sector of the Lee-Wick Standard Model
has also been constrained in Ref. [7].

There has been at least one attempt [8] to use the concepts of these Lee-Wick
constructions in cosmology where the authors were able to show the bouncing
nature of the universe whose energy is dominated by the energies of a scalar
field and its Lee-Wick partner. In Ref. [9] the authors tried to formulate a
possible thermodynamic theory of particles which includes the Lee-Wick
partners using a method of statistical field theory previously formulated by
Dashen, Ma and Bernstein in Ref. [10].

The present article mainly focusses on a variant of the original Lee-Wick
idea,222Some work in this direction was started much before by Pauli in Ref.
[11] which was concerned about the taming of the divergences in a quantum
field theory. In 1984 Boulware and Gross [12] tried to show that the original
proposal of Lee and Wick was related to a complex implementation of the Pauli-
Villars regularization scheme [13]. The complexity of the idea arises from the
fact that the Pauli-Villars regulator fields in the Lee-Wick theories are not
just ad hoc regulator fields, they also have dynamics. To implement the Pauli-
Villars idea Lee and Wick introduced massive partner fields for all the normal
fields in the theory. The scheme becomes involved when one tries to quantize
the partner fields. It turns out that the partner fields can be quantized in
two ways. In one way the norm of the states of the partner fields on the
underlying Hilbert space remains definite and in the other case the norm of
the states of the partner fields on the Hilbert space becomes indefinite. In
the former case the energy of the Lee-Wick partner fields turns out to be
negative and in the later case the energy of the Lee-Wick partner fields
remain positive.

Both the options, as stated above, have their merits and demerits. In the
first option, where the partner field states have positive definite norms but
negative energy, the theory remains quantum mechanically understandable but
does not have a proper ground state. There are run-away solutions. In the
other option, where the Lee-Wick partner field states live on an indefinite
metric Hilbert space but carry positive energy, there are zero-norm states
which can grow indefinitely. This option also gives rise to run-away
solutions. Historically Lee and Wick preferred to work with the theory defined
on an indefinite metric. The difficulties of the run-away solutions were
addressed by applying future boundary conditions which again made the theory
non-causal.

The present article deals with the option which Lee and Wick discarded, a
quantum field theory of Lee-Wick fields whose states do live on a definite
metric Hilbert space but has negative energy. The motivation for such an
unconventional work comes from a very interesting result related to the
thermodynamics of the standard Lee-Wick theory as given in Ref. [9]. It turns
out that the thermodynamics of the indefinite metric, positive energy Lee-Wick
partners is exactly the same as definite metric but negative energy Lee-Wick
fields. This similarity of the thermodynamics of the two different scenarios
gives us a glimpse of the path which Lee and Wick did not take historically.

There are a plethora of problems related with the option which is presented in
this article, the most important of them being that the theory is
energetically unstable. Presently we do not give all the pathological
properties of the alternative Lee-Wick prescription, nor do we know the cures
of all the formal (pathological) diseases of the theory. The formal aspects of
the negative energy Lee-Wick sector remains mostly open for further
investigation in the near future. Inspite of all the conceptual difficulties
related to run-away solutions the result presented is too strong to be taken
as a coincidence. Interestingly, the energy instability of the model plays an
important role in the thermodynamics of the unusual fields and indirectly
affects the results presented in this article which match surprisingly with
the thermodynamics of the positive energy Lee-Wick fields. Readers who are
purely interested in the formal aspects of the alternative Lee-Wick
prescription can go through Ref. [12] for a more lucid and formal development
of the basic ideas.

The present article is presented in the following manner. A brief introduction
on the unusual regulator fields and their properties is presented in the next
section. Section 3 discusses the technique to find out the thermal
distribution function of the regulator fields. In section 4 the energy
density, pressure and entropy density of a gas comprising of elementary
particles and their unusual field partners are calculated using the thermal
distribution functions. The last section 5 summarizes the important results
obtained in this article.

##  2 Canonical quantization of the positive energy and negative energy Lee-
Wick fields

In the paper written by Grinstein, O’Connell and Wise, [3] on Lee-Wick
standard model, the authors proposed a higher derivative field theory as the
underlying theory of nature. The quadratic kinetic terms of the normal field
theories, both bosonic and fermionic, are regained by introducing new degrees
of freedom. It turns out that the Lagrangians of the new fields have wrong
signs. In their work Grinstein, O’Connell and Wise [3] did not give any
prescription for the canonical quantization of the new fields. In a later work
by Fornal, Grinstein and Wise [9] the authors derived the thermodynamics of
the unusual new degrees of freedom using methods of statistical field theory.
In this section we first canonically quantize these new partner fields. This
exercise will immediately show that these fields do carry positive energy. In
the next step we give the prescription for obtaining negative energy fields in
the Lee-Wick paradigm.

###  2.1 Indefinite metric positive energy case

We do the quantization for the scalar field with the understanding that the
other bosons in the theory do follow the same quantization rules. If the
Lagrangian of the Lee-Wick partner of an usual scalar field be represented as
$\xi$, then according to [3], the non-interactive part of its Lagrangian is
given as

$\displaystyle{\mathcal{L}_{\xi}}=-\frac{1}{2}\partial_{\mu}\xi\partial^{\mu}\xi+\frac{1}{2}m_{\xi}^{2}\xi^{2}\,,$
(1)

where $m_{\xi}$ is the mass of the $\xi$ field. The field $\xi$ can be
expanded, in the Fourier space, in the same fashion as a standard scalar
field:

$\displaystyle\xi(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}2\epsilon({\bf
p})}}\left[a({\bf p})e^{-ip\cdot x}+\bar{a}({\bf p})e^{ip\cdot x}\right]\,,$
(2)

where $\epsilon({\bf p})=\sqrt{{\bf p}^{2}+m_{\xi}^{2}}$ is the dispersion
relation of the Lee-Wick partner excitation. The partner field can be
quantized by the following condition:

$\displaystyle[\xi(t,{\bf x})\,,\,\pi_{\xi}(t,{\bf y})]=i\delta^{3}({\bf
x-y})\,,$ (3)

where $\pi_{\xi}\equiv\delta L_{\xi}/\delta\dot{\xi}=-\dot{\xi}$. The
quantization condition for the partner fields yields the following unusual
commutation relation

$\displaystyle\left[a({\bf p})\,,\,\bar{a}({\bf q})\right]=-\delta^{3}({\bf
p}-{\bf q})\,,$ (4)

while the other commutators involving $a({\bf p})$ and $\bar{a}({\bf q})$ are
all zero. The equation above predicts that the field excitations of the
partner fields will have indefinite norm in the Hilbert space. The unusual
commutation relation between the creation and annihilation operators of the
$\xi$ field excitations is related to the negative sign of the canonically
conjugate momentum corresponding to the $\xi$ field.

If eigenstates of the number operator,

$\displaystyle N({\bf p})=-\bar{a}({\bf p})a({\bf p})\,,$ (5)

are defined by the following way

$\displaystyle N({\bf p})|n({\bf p})\rangle=n({\bf p})|n({\bf p})\rangle\,,$
(6)

where $n({\bf p})$ is a positive integer interpreted as the number of
particles with momentum ${\bf p}$ then the Hamiltonian of the field
configuration is

$\displaystyle H=\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,,$ (7)

where we have dropped the zero-point contribution in the Hamiltonian. The
important point to note is the negative sign in the definition of the the
number operator. In this case if we stick to the conventional definition of
the number operator (the same operator without the negative sign) then it can
be shown that it will have negative eigenvalues. The quantity $-\bar{a}({\bf
p})a({\bf p})$ has a positive spectrum. This unconventional behavior of the
number operator originates from the unconventional commutation relation of the
creation and annihilation operators of the field excitations as given in Eq.
(4). The negative sign of the original Lagrangian of the $\xi$ field is
balanced by the negative sign of the number operator and consequently the
Hamiltonian of the field turns out to be positive.

For the fermionic case one can take the Lagrangian of the Lee-Wick partner
field to be

$\displaystyle{\mathcal{L}}=-{\psi}^{\dagger}\gamma^{0}(i{\hbox
to0.0pt{/\hss}\partial}-m_{\psi})\psi\,,$ (8)

where $m_{\psi}$ is the mass of $\psi$ field excitations333The interested
reader can consult Refs. [14, 15] related to fermions leaving in an indefinite
metric. The negative sign of the Lagrangian in Eq. (8) gives the unusual sign
in conjugate momentum. The fermionic field can be expanded in the Fourier
basis as

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}2\epsilon({\bf
p})}}\sum_{s=1,2}\left[a_{s}({\bf p})u_{s}({\bf p})e^{-ip\cdot
x}+\bar{b}_{s}({\bf p})v_{s}({\bf p})e^{ip\cdot x}\right]\,,$ (9)

where $a_{s}({\bf p})$ and $b_{s}({\bf p})$ are the annihilation operators of
the fermionic and anti-fermionic excitations of the $\psi$ field. Quantizing
the fermion field $\psi$ in the conventional sense

$\displaystyle\left\\{\psi(t,{\bf x}),\pi_{\psi}(t,{\bf
y})\right\\}=\delta^{3}({\bf x-y})\,,$ (10)

one gets

$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=-\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf k}\right)\,.$
(11)

In the fermionic case one can define the number operator for the fermions and
anti-fermions exactly in the same way as given in Eq. (6) and the form of the
Hamiltonian will be similar to that in Eq. (7). In this case also the Lee-Wick
excitations will have positive energy but indefinite norm. An excellent
exposition of the relationship between the unusual commutation relations of
the creation and annihilation operators (of the bosonic/fermionic fields) and
the indefinite metric it induces in the Hilbert space is presented in the
first two appendices of Ref. [2].

In Ref. [9] the authors tried to envisage the Lee-Wick partners as
intermediate resonances. According to them the normal scalar particles scatter
with each other as $\phi({\bf p}_{1})+\phi({\bf p}_{2})\to\phi({\bf
p^{\prime}}_{1})+\phi({\bf p^{\prime}}_{2})$ through two Lee-Wick resonances.
The scattering in reality happens like $\phi({\bf p}_{1})+\phi({\bf
p}_{2})\to\xi(\tilde{{\bf p}}_{1})+\xi(\tilde{{\bf p}}_{2})$ and then
$\xi(\tilde{{\bf p}}_{1})+\xi(\tilde{{\bf p}}_{2})\to\phi({\bf
p^{\prime}}_{1})+\phi({\bf p^{\prime}}_{2})$ where $\xi(\tilde{{\bf p}}_{1})$
and $\xi(\tilde{{\bf p}}_{2})$ stands for the Lee-Wick fields. In this
formalism the Lee-Wick partners are unstable resonances, with a negative decay
width, and their existence is ephemeral. Writing the $S$-matrix as
$S=1-i\mathcal{T}$ one can write the $\mathcal{T}$ matrix amplitude for
$\phi({\bf p}_{1})+\phi({\bf p}_{2})\to\xi(\tilde{{\bf
p}}_{1})+\xi(\tilde{{\bf p}}_{2})$ as

$\displaystyle\langle\tilde{{\bf p}_{1}},\tilde{{\bf
p}_{2}}|\mathcal{T}(E)|{\bf p}_{1},{\bf
p}_{2}\rangle=2\pi\delta(E_{1}+E_{2}-\epsilon_{1}-\epsilon_{2})\delta^{3}({\bf
p}_{1}+{\bf p}_{2}-\tilde{{\bf p}_{1}}-\tilde{{\bf p}_{2}})\mathcal{M}(E)\,,$
(12)

where the energy $E=E_{1}+E_{2}$ and momentum ${\bf p}={\bf p}_{1}+{\bf
p}_{2}$ are such that $(p_{1}+p_{2})^{2}=m^{2}_{\xi}$ which sets the threshold
value for the creation of the Lee-Wick resonances. Here $\epsilon_{i}$ stands
for the energy of the Lee-Wick fields and $m_{\xi}$ is the mass of the Lee-
Wick excitations. The important ingredient in Ref. [9] lies in the
prescription used for writing $\mathcal{M}(E)$:

$\displaystyle\mathcal{M}(E)=-\frac{1}{2}\frac{g^{2}}{E^{2}-{\bf
p}^{2}-m^{2}_{\xi}+im_{\xi}\Gamma}\,,$ (13)

where $g$ specifies the coupling of the normal fields with the Lee-Wick
fields. The interesting part of the above prescription lies in the overall
minus sign of $\mathcal{M}(E)$ and the negative sign in front of the decay
width $\Gamma$ in the denominator. The authors of Ref. [9] then utilize a
conventional relation between the scattering matrix elements and the grand
partition function of a thermodynamic system to predict the various
thermodynamic parameters. Thermodynamics of ephemeral resonance fields with
negative decay widths is highly non-trivial but in principle one can find it
out as done in Ref. [9].

###  2.2 When the states have positive definite norm but negative energy

In the present case we will try to find out the thermodynamics of the negative
energy Lee-Wick fields which live in a normal Hilbert space. To do this one
can start from the same Lagrangian for the real scalars as used in Ref. [3]
and as given in Eq. (1). The field expansion of the $\xi$ field can be kept
exactly the same as in Eq. (2). To have a definite metric starting with a
negative Lagrangian one has to quantize the fields with the wrong sign as

$\displaystyle[\xi(t,{\bf x})\,,\,\pi_{\xi}(t,{\bf y})]=-i\delta^{3}({\bf
x-y})\,.$ (14)

This quantization condition yields the usual commutation relation between the
the creation and annihilation operators of the $\xi$ field excitations. The
$\xi$ field excitations will now lie in a normal Hilbert space (i.e. the
states having definite norm). The price one has to pay for a cross over from
the indefinite norm landscape to the positive norm landscape is the ground
state. If we stick to the definition of the number operator as defined in Eq.
(5) then $N({\bf p})$ will have negative eigenvalues. Consequently the
Hamiltonian of the field configuration can still be written as

$\displaystyle H=\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,,$

but unlike the previous case the Hamiltonian will be having negative
eigenvalues. Although this Hamiltonian will have zero as an eigenvalue, but
unlike the indefinite norm case, this eigenvalue is not the least but the
maximum of the eigenspectrum. Consequently the definite norm Lee-Wick field
excitations do not have a proper ground state. If we interpret Eq. (5), for
the indefinite metric case, as the sum of positive energy excitations then Eq.
(5) has to be interpreted as a sum of positive energy de-excitations of the
definite metric field.

From a very similar analysis it can be shown that if one changes the
quantization condition for the fermions from that given in Eq. (10) to

$\displaystyle\left\\{\psi(t,{\bf x}),\pi_{\psi}(t,{\bf
y})\right\\}=-i\delta^{3}({\bf x-y})\,,$ (15)

and defines the number operator for the particles as $N({\bf p})=\bar{a}({\bf
p})a({\bf p})$, which has positive eigenvalues, one gets a Hamiltonian as

$\displaystyle H=-\int\epsilon({\bf p})N({\bf p})\,d^{3}p\,.$ (16)

The negative sign in the Hamiltonian arises because the Lagrangian of the
fermionic field as given in Eq. (8) carries a negative sign. Similarly the
anti-fermions also carry negative energy. In this article we are following the
convention of Lee and Wick as given in the first appendix in Ref. [2]. In
their convention the fermionic algebra can be set by a set of two
anticommutation relations (one for the positive definite norm, the other for
the indefinite norm) for the creation and annihilation operators of the
fermions and correspondingly there will be two expressions of the number
operator. According to Lee and Wick a fermionic system can be quantized by the
following set of rules:

$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=\pm\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf
k}\right)\,,\,\,\,N({\bf p})=\pm\bar{a}_{s}({\bf p})\,a_{s}({\bf p})\,.$ (17)

When dealing with the indefinite metric theory we chose the minus sign in the
anticommutator and the minus sign in the number operator. On the other hand
when we are dealing with fermion fields whose states live in a normal Hilbert
space we choose the positive sign of the anticommutator and the positive
signed number operator.

##  3 Thermal distribution functions of the regulator fields

If some one assumes the existence of such exotic negative energy Lee-Wick
fields, may be during the earliest phases of the universe, then one can
calculate the thermodynamics of the negative energy Lee-Wick fields. The
calculation of the thermodynamics of the negative energy fields starts with
the prediction of a thermal distribution function of such fields. In this
section the thermal distribution function of both the bosonic and the
fermionic Lee-Wick partners are calculated.

For the bosonic excitations of the negative energy Lee-Wick fields we know
that the number operator and Hamiltonian has the form as given in Eq. (5) and
Eq. (7). In the present case the Hamiltonian of the fields as given in Eq. (7)
has negative eigenvalues because the fields are assumed to be quantized with
the wrong sign as in Eq. (14). Due to the presence of on-shell excitations the
thermal vacuum becomes $|\Omega\rangle\equiv|n({\bf p_{1}}),n({\bf
p_{2}}),\cdots\rangle$ where $n({\bf p_{1}})$ is the number of excitations
carrying momentum ${\bf p_{1}}$. The action of the number operator on such a
vacuum is

$\displaystyle N({\bf p})|\Omega\rangle=n({\bf p})|\Omega\rangle\,.$ (18)

As the value of $n({\bf p})$ actually turns out to be negative for the kind of
Lee-Wick theory we are considering so in reality $|n({\bf p})|$ gives the
number of particles with momentum ${\bf p}$ and energy $\epsilon({\bf p})$
which are missing from the thermal vacuum. In this analysis we will consider a
non-interacting real scalar field for which the chemical potential $\mu=0$. In
the present case with a Hamiltonian of the form as given in Eq. (7), where the
number operator has negative eigenvalues, the single particle partition
function will be,

$\displaystyle z_{\rm LW}^{B}={\rm Tr}e^{-\beta H}=\sum_{|n({\bf
p})|=0}^{\infty}e^{\beta\left|n({\bf p})\right|\epsilon(\bf p)},$ (19)

where $\beta=\frac{1}{T}$. From the last equation it is seen that the series
representing the single particle partition function for the Lee-Wick partner
of a Standard model boson does not converge for $\beta>0$. Consequently we
regularize the last expression by cutting off the summation for a finite value
of $n({\bf p})$ as:

$\displaystyle z_{\rm LW}^{B}=\sum_{|n({\bf p})|=0}^{M-1}e^{\beta\left|n({\bf
p})\right|\epsilon({\bf p})}=\frac{1-e^{\beta\epsilon({\bf
p})M}}{1-e^{\beta\varepsilon({\bf p})}}\,,$ (20)

where $M$ is a dimensionless cut-off which can be made indefinitely big at the
end of the calculation.

Next we calculate the thermal distribution function of the field excitations
from the expression of the single cell partition function of the field
excitations as given in Eq. (20). In conventional statistical mechanics we can
find the single cell distribution function via

$\displaystyle f({\bf p})=\frac{1}{\beta}\left(\frac{\partial\ln
z}{\partial\mu}\right)_{V,\beta}\,,$ (21)

where $\mu$ is an auxiliary chemical potential whose exact nature is not
important for our purpose. In presence of an auxiliary chemical potential the
single particle partition function can be written as

$\displaystyle z_{\rm LW}^{B}=\sum_{|n({\bf p})|=0}^{M-1}e^{\beta\left|n({\bf
p})\right|\\{\epsilon({\bf p})-\mu\\}}=\frac{1-e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}M}}{1-e^{\beta\left\\{\epsilon({\bf p})-\mu\right\\}}}\,.$
(22)

Applying conventional methods, the distribution function can also be written
as

$\displaystyle f_{\rm B}({\bf p})=\frac{1}{\beta}\left(\frac{\partial\ln
z^{B}_{\rm LW}}{\partial\mu}\right)_{V,\beta}\,,$ (23)

which comes out to be,

$\displaystyle f_{\rm B}({\bf p})=-\frac{e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}}{1-e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}}+M\frac{e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}M}}{1-e^{\beta\left\\{\epsilon({\bf p})-\mu\right\\}M}}\,.$
(24)

Now setting the auxiliary chemical potential to be zero we get the
distribution function of the fields as:

$\displaystyle f_{\rm B}({\bf p})=-\frac{1}{e^{-\beta\epsilon({\bf
p})}-1}+\frac{M}{e^{-\beta\epsilon({\bf p})M}-1}\,.$ (25)

Figure 1: The plot of the distribution function as given in Eq. (25). The
topmost curve is for a normal Bose-Einstein distribution and the lower two
curves correspond for $f_{\rm B}({\bf p})$ for the ultraviolet cutoff $M=100$
and $200$. The inverse temperature in all the cases is $.01{\rm GeV}^{-1}$ and
the energy $\epsilon({\bf p})$ is in GeV.

This is the distribution function of the fields whose Lagrangian is as given
in Eq. (1). These fields are quantized via Eq. (14). These are not the fields
which appear in the Standard model of particle physics. The distribution
function as plotted in Fig. 1 shows that the average excitation per energy
level is negative definite. Obviously these systems describe a physical theory
which is non-trivial and the negative sign of the distribution is only
meaningful when compared with the positive definite distribution of the normal
Standard model bosons. In general these kind of distributions will produce
negative energy density and pressure but once these energy density and
pressure is added with the positive energy density and pressure of the
Standard model bosons we get a net positive energy density and pressure. The
important point to notice about the distributions is that there is no pile up
of quantas near $\epsilon({\bf p})=0$ as is in the case of the Bose-Einstein
distribution. The reason being that the spectra of the Lee-Wick excitations
with negative energy, as given in Eq. (25), has two infinite spikes as the
energy tends to zero and they cancel each other near the origin.

If we take the Lagrangian of the fermionic fields as given in Eq. (8) and
quantize them via Eq. (15) then the anticommutators of the creation and the
annihilation operators, which define the fermionic excitations of the $\psi$
field discussed in the last section, is given as

$\displaystyle a_{s}^{2}({\bf p})=\bar{a}_{s}^{2}({\bf
p})=0\,,\,\,\,\,\left\\{a_{s}({\bf p})\,,\,\bar{a}_{s^{\prime}}({\bf
k})\right\\}=\delta_{s,s^{\prime}}\delta^{3}\left({\bf p}-{\bf k}\right)\,,$
(26)

where $s$, $s^{\prime}$ may be some internal quantum numbers. We can proceed
in a similar way as done before and calculate the thermal distribution of
these excitations. If we use the conventional number operator $N({\bf
p})\equiv\bar{a_{s}}({\bf p})a_{s}({\bf p})$, which has positive eigenvalues
as 0 and 1, then the Hamiltonian of a single oscillator is

$\displaystyle H({\bf p})=-\frac{1}{2}\epsilon({\bf p})\left[\bar{a}_{s}({\bf
p}){a}_{s}({\bf p})-{a}_{s}({\bf p})\bar{a}_{s}({\bf p})\right]=-\epsilon({\bf
p})\left[N({\bf p})-\frac{1}{2}\,\delta^{3}({\bf 0})\right]\,,$ (27)

where $\frac{1}{2}\,\epsilon({\bf p})\,\delta^{3}({\bf 0})$ is the zero point
energy. The negative sign of the Hamiltonian is due to the negative sign of
the Lagrangian of the $\psi$ field. A similar analysis can be done for the
anti-fermions also. If we use this oscillator Hamiltonian to calculate the
single particle partition function there will be no problem related to the
convergence of the series. The single particle partition function for the
anticommuting fields turns out to be

$\displaystyle z_{\rm LW}^{F}=\sum_{n({\bf p})=0}^{1}e^{\beta n({\bf
p})\\{\epsilon({\bf p})-\mu\\}}=1+e^{\beta\left\\{\epsilon({\bf
p})-\mu\right\\}}\,,$ (28)

where $\mu$ is an auxiliary chemical potential. Now applying the formula in
Eq. (23) and setting $\mu=0$ at the end we get the distribution function for
the definite metric Lee-Wick partners of the Standard model fermions as:

$\displaystyle f_{\rm F}({\bf p})=-\frac{1}{e^{-\beta\epsilon({\bf p})}+1}\,.$
(29)

Figure 2: The plot of the distribution function as given in Eq. (29). The
topmost curve is for a normal Fermi-Dirac distribution at $\beta=.01{\rm
GeV}^{-1}$ and the lower two curves correspond for $f_{\rm F}({\bf p})$ for
$\beta$ values $.01{\rm GeV}^{-1}$ and $.1{\rm GeV}^{-1}$ and the energy
$\epsilon{({\bf p})}$ is in GeV.

Unlike the previous case, in the present scenario the distribution function
has no dependence on the dimensionless regulator $M$.

The negative sign of the distribution function signifies that the present
field configurations arises due to a de-excitation or loss of positive energy
particles. The vacuum defined is not stable and there exists much less
energetic states than the vacuum itself. These kind of fields are unstable.
The maximum energy of the field configurations is zero.

##  4 Energy density, pressure and entropy density from the distribution
function.

###  4.1 The bosonic case

To calculate the relevant thermodynamic quantities for the bosonic field from
a statistical mechanical point of view we will employ Eq. (25). The energy
density can be calculated using the following known equation

$\displaystyle\rho=\frac{g}{\left(2\pi\right)^{3}}\int\epsilon({\bf p})f_{\rm
B}(\epsilon)d^{3}p=\frac{g}{2\pi^{2}}\int_{0}^{\infty}\epsilon({\bf p})f_{\rm
B}(\epsilon)|{\bf p}|^{2}d{|\bf p}|\,,$ (30)

where $g$ stands for any intrinsic degree of freedom of the particle. For a
relativistic excitation $\epsilon^{2}={\bf p}^{2}+m^{2}$ where $m$ is the mass
of the bosonic excitations. Changing the integration variable from $|{\bf p}|$
to $\epsilon$ one gets

$\displaystyle\rho=-\frac{g}{2\pi^{2}}\int_{0}^{\infty}\left(\epsilon^{3}-\frac{m^{2}}{2}\epsilon\right)\frac{d\epsilon}{e^{-\beta\epsilon}-1}+\frac{Mg}{2\pi^{2}}\int_{0}^{\infty}\left(\epsilon^{3}-\frac{m^{2}}{2}\epsilon\right)\frac{d\varepsilon}{e^{-\beta\epsilon
M}-1}\,.$ (31)

In the above integral it is assumed that $|{\bf p}|\gg m$ and to have a closed
integral the lower limit of the integral is assumed to be zero. In the extreme
relativistic limit the system temperature $T\gg m$. Both of the integrals can
only be done when $\beta<0$, and in that case the result of the last integral
is

$\displaystyle\rho=-\frac{g}{2\pi^{2}}\left(\frac{\pi^{4}T^{4}}{15}-\frac{m^{2}\pi^{2}T^{2}}{12}\right)+\frac{g}{2\pi^{2}}\left(\frac{\pi^{4}T^{4}}{15M^{3}}-\frac{m^{2}\pi^{2}T^{2}}{12M}\right)\,.$
(32)

Analytically continuing the above result for $\beta>0$ and taking $M\to\infty$
we see that for normal temperatures the energy density for extreme
relativistic excitations of the bosonic fields is of the following form

$\displaystyle\rho=-g\left(\frac{\pi^{2}T^{4}}{30}-\frac{m^{2}T^{2}}{24}\right)\,.$
(33)

As expected, the energy density turns out to be negative for the excitations
in this case. The pressure of the bosonic field excitations can be found out
from

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{g}{\left(2\pi\right)^{3}}\int\frac{|{\bf
p}|^{2}}{3\epsilon}f_{\rm
B}(\epsilon)d^{3}p=\frac{g}{2\pi^{2}}\int_{0}^{\infty}\frac{|{\bf
p}|^{4}}{3\epsilon}f_{\rm B}(\varepsilon)d|{\bf p}|\,.$ (34)

Following similar steps as in the case of the energy density, it is seen that
the pressure of extremely relativistic excitations of the bosonic fields turns
out to be

$\displaystyle
p=-g\left(\frac{\pi^{2}T^{4}}{90}-\frac{m^{2}T^{2}}{24}\right)\,.$ (35)

The entropy density of the bosonic field is simply given by

$\displaystyle
s=\frac{\rho+p}{T}=-g\left(\frac{2\pi^{2}T^{3}}{45}-\frac{m^{2}T}{12}\right)\,.$
(36)

These values of the energy density, pressure and entropy density exactly match
the corresponding values calculated for the conventional, positive energy Lee-
Wick partners. In [9] the authors were trying to formulate thermodynamics for
a higher-derivative theory. The higher derivative theory was converted into
standard theory (theory up to a second derivative) with the introduction of
Lee-Wick partners whose states have indefinite norms. The authors in the
previous work did not quantize the system explicitly but were working with the
form of the propagators of the Lee-Wick partners.

If we assume that in the early universe for each bosonic degrees of freedom in
the Standard model there exist a corresponding Lee-Wick bosonic degree of
freedom whose field configuration has negative energy then the net energy
density, pressure and entropy density of the early universe turns out to be

$\displaystyle\rho_{B}=\rho_{\rm
SM}+\rho=\frac{gm^{2}T^{2}}{24}\,,\,\,\,\,p_{B}=p_{\rm
SM}+p=\frac{gm^{2}T^{2}}{24}\,,\,\,\,\,s_{B}=s_{\rm
SM}+s=\frac{gm^{2}T}{12}\,,$ (37)

which are all positive as expected. Here the energy density, pressure and
entropy density for the Standard model bosonic particles are $\rho_{\rm
SM}=g\frac{\pi^{2}T^{4}}{30}$, $p_{\rm SM}=g\frac{\pi^{2}T^{4}}{90}$ and
$s_{\rm SM}=g\frac{2\pi^{2}T^{3}}{45}$ respectively [16].

###  4.2 The fermionic case

In this subsection we apply Eq. (29) to find the energy density, pressure and
entropy density of the fermionic excitations. In this case the distribution
function do not have any dependence on the regulator $M$. For relativistic
excitations the integrals which give the energy density and pressure for the
fermionic case are exactly similar with the bosonic case except that now we
have to use the distribution for the fermions. The integrals can be easily
done, granted $\beta<0$, but the results can be analytically continued for
positive temperatures. The results in this case are listed below. The energy
density, pressure and entropy density of the Lee-Wick partners are as follows:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{4}}{240}-\frac{m^{2}T^{2}}{48}\right)\,,$
(38) $\displaystyle p$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{4}}{720}-\frac{m^{2}T^{2}}{48}\right)\,,$
(39) $\displaystyle s$ $\displaystyle=$
$\displaystyle-g\left(\frac{7\pi^{2}T^{3}}{180}-\frac{m^{2}}{24}\right)\,.$
(40)

The energy density and pressure quoted above are equivalent to the energy
density and pressure for the positive energy Lee-Wick partners as calculated
in [9] for the special case of $g=2$. If we assume that to each unusual
fermionic degree of freedom there corresponds one standard fermionic degree
from the Standard model, then the total fermionic contribution is

$\displaystyle\rho_{F}=\rho_{\rm
SM}+\rho=\frac{gm^{2}T^{2}}{48}\,,\,\,\,p_{F}=p_{\rm
SM}+p=\frac{gm^{2}T^{2}}{48}\,,\,\,\,s_{F}=s_{\rm SM}+s=\frac{gm^{2}T}{24}\,$
(41)

which are all positive. Here the energy density, pressure density and entropy
density for the Standard model fermionic particles are $\rho_{\rm
SM}=g\frac{7\pi^{2}T^{4}}{240}$, $p_{\rm SM}=g\frac{7\pi^{2}T^{4}}{720}$ and
$s_{\rm SM}=g\frac{7\pi^{2}T^{3}}{180}$ respectively [16].

It is worth pointing out here that there is some confusion regarding higher
derivative theories of fermions. The confusion is regarding the number of Lee-
Wick partners (one left-handed and the other right-handed) of the chiral
fermions. The authors of Ref. [9] claim that there will be two positive energy
Lee-Wick partners of a chiral fermion which are interrelated. Where as in Ref.
[17] the author claims that the two positive energy Lee-Wick partners of the
chiral fermion may not be interdependent. In that case the Lee-Wick degrees of
freedom exceeds the one of its Standard model partner yielding negative
energy, pressure and entropy density. This issue is yet to be resolved.

##  5 Discussion and conclusion

Initially it was pointed out that Lee-Wick’s idea of implementing the Pauli-
Villars regularization scheme can be implemented in two ways. This idea was
presented by Boulware and Gross [12] way back in 1984\. In one way the
regulator fields live in a indefinite metric space but carry positive energy
and in the other way the regulator fields live in a definite metric space but
carry negative energy. Lee and Wick took the first option and tried to redress
the issue of indefinite norm in such a way that unitarity is preserved in the
theory. The second option remained uncultivated. In this article we explored
the second option with limited means. No cures for the energy instability of
these kind of theories are known to the present authors. The results presented
in the article show the dubious nature of the energy instability of the
fields, but instead of making the theory meaningless the same instabilities
conspire to produce a result which matches with the thermodynamics of the Lee-
Wick partners living in the indefinite metric space.

In this article we have studied a system of bosonic and fermionic fields,
whose Lagrangians have the wrong sign and, which are quantized with the wrong
sign of the commutators and the anticommutators. These fields are Lee-Wick
partners who live in a normal Hilbert space but have negative energy
excitations. The negative energy of the field configuration is not due to any
particular form of the potential but solely an outcome of the negative sign of
the Lagrangian and the modified quantization process. The vacuum of the theory
is not the state with the lowest energy, it is rather the state with the
maximal energy making the field configuration unstable. The bosonic and
fermionic degrees of freedom do still follow commutation and anticommutation
relations and specifically the fermionic fields still follow the Pauli
exclusion principle. In this article the emphasize had been on the calculation
of energy density, pressure and entropy density of the unusual field
configurations.

To calculate the above mentioned thermodynamic quantities one requires to have
a statistical mechanics of the field excitations. One encounters the
difficulty of a diverging sum when calculating the single particle partition
function of the bosonic fields. Keeping to conventional ways, where the
temperature of the system is positive definite, the partition function can
only be summed when one uses an ultraviolet cutoff. The distribution function
calculated from the partition function turns out to be negative definite,
which is a nontrivial result. The negative nature of the distribution function
implies that there must be an average loss of particles in any energy level.

The energy density, pressure calculated from the distribution functions of the
unusual fields discussed in this article match exactly with the results
calculated by Fornal, Grinstein and Wise in [9]. The derivation of the new
distribution functions and the connection between the regulator field
thermodynamics as presented in this article and the thermodynamics of the Lee-
Wick partners, as presented in Ref. [9], is one of the main motivations for
this work. The main emphasize of the present article has not been to
recalculate the results obtained in Ref. [9] as the theory presented in this
article is not the same as that of Ref. [9]. The two theories are quantized
differently. The similarity of the thermodynamic results of the two variants
of the Lee-Wick model implies that the different kind of instabilities
plaguing the theories are related in presence of a thermal bath. From the main
analysis of this article it can be inferred that the conventional Lee-Wick
prescription is equivalent to its variant in the thermodynamic sector. The
analysis of the thermodynamics of the positive definite metric Lee-Wick
partners gives a clear idea about the origin of the negative energy density
and pressure of these field configurations. The theory presented in the
article is amenable to the standard techniques of finite temperature field
theory. One can utilize the results of finite-temperature quantum field theory
to calculate various thermal effects, like the thermal mass of the Standard
model particles in presence of the thermalized Lee-Wick partners, in the early
universe. The thermal distribution functions of the Lee-Wick partners
calculated in the present article can be used to write the propagators of the
unusual fields in the real-time formalism [18].

The fact that the thermodynamics of the indefinite metric, positive energy
Lee-Wick fields and the thermodynamics of the definite metric but negative
energy Lee-Wick partners turns out to be the same remains an interesting
result which invites further work in these fields in the future.
Acknowledgement: We want to thank Subhendra Mohanty for pointing out some
subtle ideas involved in this article.

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